Why does spacetime propagate gravitational waves? Setting the scene:
If I drop a stone into water, the stone will create a depression in the water that the surface tension of the water and gravity (so says the Wiki article: Dispersion (water waves)) will work to neutralise so that the water surface again becomes flat. This has the side effect of creating a disturbance in the "flatness" of the water surface that actually moves (away from its origin).
If I drop a stone in sand, which is also composed of particles, the depression largely remains when I remove the stone, and no traveling waves were created.
Back to the question:
Gravitational waves apparently travel through space, so in my primitive understanding of things, space (or spacetime?) must have a neutral "shape" that it is compelled to resume after being deformed. Is this true or am I misunderstanding?
If it's true, why doesn't spacetime behave like the sand in that it preserves deformations after the object that caused them has moved away?
Update 1:
Thank you for the replies! I think Cleonis' extended reply saying that gravitational waves are predicted to exist (or actually necessarily exist if I've understood correctly) and propagate in spacetime when assuming the Einstein equivalence principle is a satisfying answer if one is not to try to explain what the so-called "fabric of spacetime" is actually made of. I haven't done a textbook study of General Relativity though, so it's (yet) beyond me how one gets from the equivalence principle to gravitational waves. Maybe there's a non-mathematical way to explain this? (as hand-wavy as needed) [Update 2: See the 2nd reply by Cleonis]
I gather that the question of what the fabric of spacetime physically is is yet to be solved.
Cleonis and Ed999 inspired me to do some googling.
"spacetime as a fluid": This in particular leads to a Wiki article "Superfluid vacuum theory" in which the physical vacuum is assumed to be a superfluid (a fluid with zero viscosity) and consequences are then derived. It seems to be a fringe idea though.
"spacetime elasticity": This yielded two small technical papers. "What is the Stiffness of Spacetime?" (http://physics.princeton.edu/~mcdonald/examples/stiffness.pdf) and "The Elasticity of Quantum Spacetime Fabric" (http://www.bio21.bas.bg/conference/Conference_files/sa17/Cartas.pdf). The latter makes the intriguing claim that: 
"The conclusion of all these is that the spacetime has to have a
small-scale structure, which is granular. The “atoms” of the space
time have to be subjects of a peculiar force. From the elastic point
of view, there are good arguments for the Casimir force to qualify as
the necessary force."
But the paper was too technical for me to make out if the author actually makes a convincing case for this.
Update 3:
Ed999 (in a new reply) gives an exposition of the idea of thinking of a gravitational wave as a compression wave in a fluid of spacetime "granules" expanding radially from a source so that the wavefront of the traveling disturbance forms an expanding spherical shell.
Two sections had me wondering, though.
Quoting from the reply:
"The strength of the wave is one-quarter at distance 2x, compared to distance x, simply because at distance 2x the expanding nature of the wave (i.e. its spherical expansion pattern) means that each unit of energy must displace four times as many of the granular units of spacetime"
and
"... the tensor pushes against the next adjacent unit, forcing it in the opposite direction; but, like a tiny spring, it also recoils after doing so, returning to a stationary state. Hence the deformation is temporary, i.e. elastic..."
Wouldn't the wavefront gradually lose its speed (even with elastic collisions between granules) when the kinetic energy transferred between the granules is spread out over an increasing amount of granules as time passes?
He ends his reply with:
"The overall implication of the math, both Newtonian and Einsteinian, is that waves of gravitation and electromagnetism obey the same physical laws, and for the same reasons: that both are a wave motion in a granular medium; a medium which responds to the ordinary, well-understood geometric principles associated with a spherical type of wave propagation based on vibration."
If we think of a traveling electromagnetic disturbance as also being a compression wave in a granular medium, I'm having some trouble imagining how a highly localized disturbance traveling in a straight line can arise such as when an atom emits a photon.
It makes some sense to me if I imagine the photon as being propagated as a single moving granule (or a group traveling in the same direction), but it seems then that the photon should change its traveling direction every time the carrier granule hits another granule at an odd angle, and just generally lose its speed as its kinetic energy gets spread out over an increasing number of granules through collisions along the way. Maybe this would correspond to an increasing wavelength of the photon over time?
Update 4:
This question from 2015 asks essentially the same thing as my post:
How do gravitational waves work without internal tension?
The answers to that are not restrained in their technicality as is the case here.
 A: Indeed it is assumed that in the absence of a source of gravitation spacetime will be uniform. In that sense we can say that spacetime curvature must be regarded as a form of elastic deformation. As in: when the source of deformation goes away, the deformation goes away.
Other than that the nature of spacetime is unknown. This elastic property must be granted in order to formulate a theory at all.
More generally:
In terms of GR very little is known about the nature of spacetime. In order to formulate GR at all some assumptions must be made. For sure: the fact that such a succesful theory has been constructed is strong supporting evidence for these assumptions. Our concept of the nature of spacetime is limited to the content of those assumptions.

[LATER EDIT]
In response to the comment by Ben Crowell:
I will first present this response, then I will explain why I judged this lengthy response to be necessary.
In the original question Mads asked: "Why doesn't spacetime preserve deformations after the object that caused the has moved away?"
I infer that Mads is asking: "Why does spacetime undergo elastic deformation rather than  plastic deformation?"
Of course, plastic deformation of spacetime is such a bizarre concept that it just never occurs to a physicist to consider it, it doesn't enter the mind.
But of course we can interpret the question in a more general sense: "what is the underlying physics that makes spacetime behave in the way that it does?"
In the following the background is the distinction between 'implicit' and 'explicit'
As we know: Einstein used Einstein's principle of equivalence and the demand that in non-relativistic conditions GR must reproduce Newton's inverse square law of gravity as things that his new theory would have to satisfy in order to be considered viable. By and large these two contraints were sufficient to narrow down the possibilities to the 1915 GR.
Implicitly the principle of equivalence has vast consequences.
If you grant these two:
- the principle of equivalence leads to GR
- GR implies the existence of gravitational waves
Then it follows that you must grant the combination: principle of equivalence ultimately implies the existence of gravitational waves.
While the logical implications are far reaching, the principle of equivalence does not describe spacetime explicitly. In that sense the principle of equivalence is very cautious. The principle of equivalence is minimal; it states the very minimum that is needed in order to narrow down the GR search sufficiently.
Let me make a comparison: historically the second law of thermodynamics was first an empirical law. Later on the science of statistical mechanics moved the description of entropy to a deeper level. Before the development of statistical mechanics the pre-statistical thermodynamics was a completely succesful theory. Still, at a later time a framework was developed that established a foundation underneath its predecessor. 
We cannot exclude the possibility that at some point in the future a theory of spacetime will be developeded that is to GR what statistical mechanics was to pre-statistical thermodynamics. If such a theory of spacetime is developed it may describe on a fundamental level a physical mechanism allows spacetime to curve, and that causes it to return to uncurved state when the source of gravitation moves away again.
To Mads, who asked the original question:
by conscious decision the assumptions that are granted in order to formulate GR are minimal. We don't know why spacetime has the properties that it has. What we know is what we need in order to be able to formulate GR; the principle of equivalence. We grant that. Because of the success of GR we judge this to be a good call.
Now: why such a lengthy response, far longer than my first reply to Mads.
To Ben Crowell: 
1) I urge to you always apply historical awareness.
Every new theory of physics is formulated by starting from one or more assumptions that must be granted in order to formulate the theory at all. Example: on publication of the Principia many of Newton's contemporaries objected that since no physical mechanism for the inverse square law was described the theory should be rejected. Newton argued that the very success of the inverse square law of gravitation was sufficient evidence in itself. Newton was right of course. 
In the history of physics it is never the case that the reigning theory is the exhaustive theory that will not have a successor. A new theory replaces a predecessor if it moves the description of the physics taking place to a deeper level of description, thus deepening the understanding. 
Yes, of course GR is a thoroughly successful theory of motion and gravitation. But like all theories of physics in order to be able to formulate it at all some assumptions are granted. Granting these assumptions is justified by the success of the theory.
2) I urge you to be always aware of far reaching implications of starting assumptions. Yes, of course GR does not make explicit statements about elastic properties of spacetime. But if you grant that propagation of gravitational waves is possible you have implicitly granted the elastic property that enables that. 
Here on stackexchange comments are limited to 600 characters for a good reason; the stackexchange designers want to give some room for comment, but everybody is urged to refrain from getting sucked into protracted one-on-one debate. I support that policy wholeheartedly.
So yeah, writing this protracted response-to-a-comment feels very awkward. 
In general, if you feel an answer that I wrote is off, then please cast your response in the form of a additional answer to the question, an answer where you try to lead the reader in the direction that you deem better. Let the reader decide what answer is the best fit for the question.
A: 
Gravitational waves apparently travel through space, so in my primitive understanding of things, space (or spacetime?) must have a neutral "shape" that it is compelled to resume after being deformed. Is this true or am I misunderstanding?

I would say this is not a good way of thinking about it. For example, after a gravitational wave passes over the earth, the spacetime around the earth doesn't go to a preferred flat shape, it just goes back to the original state of curvature that it had before because of the earth's field.
A good analogy is what happens if I broadcast a radio beep. The radio signal is an electromagnetic field, and the earth also has its own static magnetic field. While the radio signal is propagating, its electric and magnetic fields just add onto the earth's magnetic field. When the radio wave is gone, the earth's magnetic field is back to what it had been.
This is called superposition, which is just a fancy way of saying addition. Another classic example of superposition is two sets of ripples on a pond spreading through each other without interacting.
Now it's not actually true that general relativity obeys a law of superposition, but it is an extremely good approximation for a small-amplitude gravitational wave passing through the static curvature of an object like the earth.
A: This is an answer to update 1 of your question.
I should point out: the design of stackexchange is that a sufficiently distinct follow-up question should be asked separately, as a question on its own.
The question:
Is there any hand-wavy path from the principle of equivalence to the existence of gravitational waves?
The case of gravitational waves is very different from electromagnetic waves. For more about that I suggest you read the article Aberration and the speed of gravity by Steven Carlip. 
Despite the differences I will still use electromagnetism as an analogy.
Around the time of Faraday the concepts of 'electric field' and 'magnetic field' were introduced, and Faraday introduced a concept of 'field lines'. The field was thought of as a mediator of the interactions, electrostatic interaction and magnetic interaction. It wasn't known whether these fields actually existed. This hypothesized field is not an observable; what we measure is how in the presence of such a hypothesized field trajectories of particles are affected. (For example the trajectories as particles move through a cloud chamber.) Whatever the case, the field concept is at least a good instruments to guide thinking about the problems. To aid his thinking, Maxwell used a physical model of the ether. This model was designed to embody known properties of electricity and magnetism. Some of the logical implications of the model proved to be true physical properties of the real world, that is how Maxwell's model was worthwile.
A necessary condition for wave propagation is capability of oscillation in whatever form. Maxwell noticed that in his model conditions for oscillation were present. There is an unstrained state, a force can stress the state of the model away from the unstrained state, and on rebound the changing state will overshoot. That is, once the state is at a rate of change, it tends to continue at that rate of change, analogous to inertia. That is what you need for oscillation. Maxwell showed that if you granted the existence of an hypothetical (but plausible) phenomenon that he named 'displacement current', then according to his theory electromagnetic waves should exist, and he could calculate the speed of propagation, and it was to within known accuracy of measurement the same as the speed of light.
The very existance of electromagnetic waves also implied that the electromagnetic field is a true physical entity, not just a mental construct. An electromagnetic wave continues to propagate, regardless of whether the source that emitted still exists. 
Maxwell's prediction of the speed of electromagnetic waves is somewhat analogous to the way that Isaac Newton calculated the speed of sound propagation from first principles. Air meets the requirements for oscillation:
- air has elasticity
- air has inertial mass, so on rebound it will overshoot
In the Principia Newton presented a calculation that used these properties: given the elasticity of air and its mass per unit of volume the speed of sound can be derived.
In the case of electromagnetism the restoring force that acts to restore the state back in the direction of unstressed state is very, very strong. Thus the speed of propagation of electromagnetic waves is very, very fast.
Principle of equivalence and gravitational waves 
As mentioned elsewhere, the principle of equivalence has vast implications. As electromagnetism grew from strength to strengh physicsts began to explore the possibility of the existence of a gravitational field, a field that acts as the mediator of gravitational interaction. The principle of equivalence implies the supposition that spacetime itself is the very gravitational field. According to GR spacetime itself is acting as the mediator of gravitational interaction.
If you grant the principle of equivalence you have implicity granted that spacetime itself has physical properties, to be described by a quantitive theory. There is an unstrained state, which is referred to as 'geometrically flat spacetime', and any source of gravitation causes a strained state of the field, that is referred to as 'curvature of spacetime'.
My knowledge of GR is insufficient to go beyond this stage of discussion.
Let me point out some stark differences with the history of Maxwell's conclusion that electromagnetic waves must exist. It was possible for Maxwell to calculate the propagation speed of electromagnetic waves from first principles. However, in the case of gravitational waves it is assumed that they propagate at the same speed as light. It is clearly very plausible that gravitational waves propagate at lightspeed, but logically there is room for alternative theories of gravitation according to which gravitational waves propagate slower. (A recent LIGO observation with gravitational wave and light from a cataclysmic event arriving effectively simultaneously is strong evidence that gravitational waves indeed propagate at lightspeed.)
According to the wikipedia page about gravitational waves Einstein went back and forth on the question as to whether gravitational waves exist, so evidently this is among the most opaque aspects of GR.
A: A ripple (wave) in a liquid, such as water, is a type of movement which the relatively fluid (no pun intended!) nature of the medium allows, because of the fact that the medium is fluid: i.e. it tends not to resist motion of the molecules, having in fact a low degree of resistance to such movement.
Sand, being a solid, has a far greater degree of resistance. There is genuine friction between the molecules of such a solid. Gas, on the other hand, exhibits almost no friction between the gas molecules at all, so is even less resistant to their movement than a liquid. Clearly, it is a matter of density: the low density of the molecules in a gas allows greatest movement of those molecules, and the high density of the molecules in solids allows least movement of those molecules.
When the medium is considered in relation to a wave, this matter of density becomes perhaps less important. We know that waves can exist in a gas: sound waves in the air are well enough understood. Equally, waves can exist in a liquid, so waves in a pond or an ocean are also quite familiar to us. But, in fact, waves exist in solids, within our normal experience: earthquake waves are pretty much well understood too.
But if we consider an earthquake, we usually find that rock - i.e. solid rock - is a dangerously effective conductor of such waves. Sand is, in contrast, more an effective dampener of such waves. So the nature of the medium is at least as important as its being solid: a solid and rigid structure, such as a rock, has quite different behaviour to a solid but non-rigid medium such as sand.
We might perhaps consider the nature of spacetime in the light of our experience of these waves in other mediums. Theoretically, the fact that electromagnetic waves do propogate in spacetime implies that the fabric of spacetime must be behaving more akin to a solid but rigid structure, one capable of transmitting a wave, and not akin to a sandy medium which tends to rapidly damp-down such a wave.
Wave-motion seems to require one molecule, in (say) a liquid, colliding with the next, thus passing on the motion, but thereby returning to rest: so the wave-front represents a sequence of collisions, causing what amounts to a compression-front to travel through the water. All wave-motion is essentially a motion in a medium, where the individual units comprising the medium each move very little distance, and pass on motion without themselves going anywhere.
Spacetime, like water or gas or rock, will have a "rest" state, which an (electromagnetic) wave only briefly disturbs before the original rest state is resumed. Then it will be briefly disturbed again, by the passage of the next wave-front. Theoretically, a gravity wave is a similar disturbance of the spacetime fabric.
Sand does not have the coherence to pass on wave-motion. Solid rock does, as does water and air. It appears to be a matter of the sand offering too much resistance, which solid rock - due to its rigid structure - does not offer. The rigity of the rock is a positive aid to conducting the earthquake wave, even though on the face of it a solid looks mighty unpromising as a wave conductor. But of course you need a mighty strong source of vibration to set up that wave effect in a solid, compared to (say) the energy required to make a sound wave in a gas.
The distiction is somewhat mirrored in spacetime: little energy is required to generate an electromagnetic wave, any star can do it (actually, on a smaller scale even I can do it, by lighting a candle); whereas to generate a gravity wave requires an initial energy disturbance that only two merging black holes can generate.
Why does sand not go back to its original state? Sand does not have the capacity to pass on energy in a wave-motion. A medium that has no capacity to transmit such a motion will never resume its initial rest state, because the effect is incomplete. It receives the motion, but does not pass it on. Newton pointed out that action and reaction are equal and opposite: if the molecule passes on the motion, as in a liquid, the act of passing it on generates an equal and opposite reaction, returning the molecule to where it began. But not if the motion is not passed on.
In theory, if you input sufficient energy, then - just as with the earthquake and the solid rock - the sand probably would exhibit wave motion. But I suspect that the amount of energy required would probably be such as to melt the sand, hence it would behave thereafter in the normal manner of a liquid, and its value for studying wave motion in solids would be moot. Your dropping a small rock doesn't yield nearly enough energy.
Sand, as such, lacks the surface tension which causes liquids to give such a vivid demonstration of the principles involved. That surface tension represents the coherence bond which holds the molecules together, one to another, that seems to be necessary to permit the liquid to transmit the wave-motion. 
A: Prodded into providing a second answer, and finding 600 characters to be too restricting in which to do so, this is my supplementary comment.
I don't like the suggestion, implied above, that Einstein was not able to explain Newton's basic assumption, namely the inverse-square law. If you take a look at Einstein's math, you could never reach that conclusion: the math speaks for itself.
I'm not sure I'm really qualified to speak for Einstein! You really must look at the equations and judge them for yourself. It's complex, and I can only try to highlight a few salient points.
It seems to me that Newton's law of gravity makes an attempt to take a single known fact, namely the actual behaviour of the gravity field, usually termed the inverse-square law, and to build an entire theory of gravitation upon that single fact. Einstein then comes along, explains what that law means, and then builds a further theory of gravitation, from the exact same starting point but introducing relativistic effects that Newton had neglected.
For the record, the inverse-square law is simply a statement of an observed fact: the strength of the gravity field falls with distance. Having made a series of detailed measurements, Newton realised that if you take any mass, when you measure the gravitational strength of it at a distance, x, if you double the distance and then measure the field's strength, at the new distance, 2x, the field strength is only one-quarter of its strength at distance x.
Newton then formulated his theory, that the field strength is inversely proportional to the distance. Specifically, that it is always one over the square of the distance. Thus at a distance of 1 million miles it is 1/1² (i.e. 1), and at 2 million miles it is 1/2² (i.e. 1/4). Thus, by doubling the distance the field's strength falls to a quarter of its former value.
Einstein's math makes it clear that he understood the relationship: it is the relationship which arises from the circumference of a circle, when projected into 3-dimensions, as a sphere: in other words, what Newton was measuring, Einstein says, is a relationship based upon the surface area of a sphere.
With a sphere, if you take a central point, such as a planet or a star, and measure its gravitational strength at a distance of 1 million miles, then double the distance and measure again, the strength falls to a quarter; but that is not the only thing which falls to a quarter: at 1 million miles, if you measure the surface area of a sphere of radius 1 million miles (in other words, the surface area of a sphere of which the planet or star is the center), you obtain a value of x for the surface area (in, say, square miles); but when you repeat that measurement, at double the distance, the surface area is exactly 4 times greater.
What Einstein had realised (perhaps even Newton, too) is that a beam of sunlight emitted by the star, which, at a distance of 1 million miles, illuminates an area of 1 square mile on the surface of our hypothetical sphere of radius 1 million miles, will, when it has travelled double the distance, be illuminating an area of 4 square miles.
This simple geometric fact tells us quite a lot: the strength of gravity falls to 1/4 if the distance is doubled, but so does the strength of sunlight. The same amount of light now illuminates 4 times the area, so each square mile is receiving only a quarter of the total. This rather striking co-incidence is not really a co-incidence at all. If we think of gravity as a wave, and light as an electro-magnetic wave, we can begin to see that they might have some common properties.
We can see, for one thing, that they are both obeying the inverse-square law. What does this imply? Well, one implication is that both gravity and electromagnetic radiation are propogating in a spherical pattern, radiating out 3-dimensionally from a central point, as a sphere (or shell) of energy.
This mechanism causes the inverse-square effect, since a given quantity of energy, q, emitted from a central point, a star, will diminish in strength with distance as predicted by Newton: if we project a set of imaginary spheres around the star, set at intervals of 1 million miles, the strength of the gravitation and of the emitted light  (measured at any two of our imaginary spheres) falls in proportion to how much the surface area of the sphere has altered. As the distance from the star exactly doubles, the surface area of the imaginary sphere exactly quadruples, and the strength of the gravity wave and the electromagnetic wave fall to exactly a quarter.
Newton's math thus gives us the striking fact that, for both types of wave, doubling the radius of the sphere, thus quadrupelling the surface area with which the wave must interact, causes the measured strength of the wave to fall to a quarter. The logic of the math is that the spacetime medium which is transmitting the wave is spreading it out over four times the surface area, and it is thereby having only one-fourth of the effect per unit of area.
The same mechanism which allows spacetime to transmit energy as a wave, also causes spacetime to curve. Logic demands that spacetime must be flexible: it cannot vibrate if it is not, and this vibration is what is permitting the energy transmission. 
Newton tells us that action and reaction are equal and opposite. What this implies is that which we would logically expect: as in an ocean, where the water molecules bump one against the next to pass on the motion which we perceive as a wave, the granules of spacetime are in collision with one another, but they do not go anywhere: they pass on the motion, but then return to their starting point. The reaction to passing on the motion is equal, and is opposite, putting them back where they started.
Elastic deformation is a result of the tensor force, which separates the individual units of spacetime, being compressed: as the cell is impacted from the direction of the centre of the sphere, i.e. the star, the tensor pushes against the next adjacent unit, forcing it in the opposite direction; but, like a tiny spring, it also recoils after doing so, returning to a stationary state. Hence the deformation is temporary, i.e. elastic (rather than plastic, i.e. permanent).
This flexibility allows a gravity wave to be passed on, or an electromagnetic wave. The strength of the wave is one-quarter at distance 2x, compared to distance x, simply because at distance 2x the expanding nature of the wave (i.e. its spherical expansion pattern) means that each unit of energy must displace four times as many of the granular units of spacetime (put another way, there are only one-fourth the number of energy units/quanta arriving per square mile of surface area).
It suggests that ordinary gravity is imposed by a more permanent deformation of the tensor. Logic suggests that gravity is most likely simply a reduction in the resistance to inertia (the force holding a particle in one place) possessed by the granular structure of spacetime. If the granules offer less resistance in one direction, a particle in motion, which follows the path of least resistance, will inevitably tend to move in that direction. 
If the tensor (by reason of the presence of a central mass) is shorter in the direction toward the star or planet, and so the distance between the granules is less in that direction, this offers a logical basis (a mathematical reason) for the particle in motion to move in that direction: where the energy requirements are lower for moving toward the central mass, as contrasted with every other direction, the particle will tend to move toward that mass. 
Where the space surrounding the star (or planet) is composed of a multitude of such granules, each having its tiny tensor(s) compressed more greatly in that direction, then to our perception, at the macro-level, it might appear as though (i.e. there could be an illusion that) space is curving, since an object of low mass injected into such a system (but with some motion/momentum of its own, and given a suitable angular momentum) might behave like it was being exposed to a curved surface. The math seems to be similar.
(The "illusion of curvature" - Logic implies that if the wave
      is expanding in a spherical pattern, then the wave-front must
      inevitably take a curved form. If we equate each point on the
      surface of our imaginary sphere, of radius 1 million miles,
      with a particular field-strength (an equal value, for inertia or
      perhaps for resistance), and view the field from that perspective,
      the pattern of the field strength must inevitably appear curved,
      as it must be uniform in strength at every point that is equidistant
      from the star.)
If we think of a particle in terms of quantum mechanics, where the granules of spacetime are closer together in one direction, the particle necessarily requires less energy to "tunnel" in that direction. Again, the math implies this effect. 
The overall implication of the math, both Newtonian and Einsteinian, is that waves of gravitation and electromagnetism obey the same physical laws, and for the same reasons: that both are a wave motion in a granular medium; a medium which responds to the ordinary, well-understood geometric principles associated with a spherical type of wave propogation based on vibration; that the tensors which bind spacetime into a cohesive whole allow the transmission of this type of energy; and that ordinary principles of motion and inertia, and of quantum tunnelling, explain gravity.
There are many implications in the foregoing for the likely composition of the granular structure of spacetime, but as they don't follow from the actual math developed by Einstein I won't complicate this discussion any further here.
