Reactionless Drives According to the third law of motion, you van't have an mass move in a particular direction unless there is a proportional opposite mass/acceleration ratio in the opposite direction.
No-one has been able to provide a convincing argument otherwise, but the best one to date is Shawyer's EM Drive. He claims some fancy relativistic effects are what allows his engine to work, but I have read some papers which claim he is a fraud.
My question is, why is it impossible to move a mass in a given direction without a proportional change in the opposite direction?
I'm not talking about a perpetual motion machine, or anything. Sure, the device would need to consume at least the amount of energy proportional to the energy required to accelerate the mass.
Here's a highly hypothetical example:
Say we either can project a gravity well in front of our vehicle, and/or project a gravity hill behind. In empty space, the effects of the gravity will be near-negligible by the time they reach any other object, however close to the vehicle they will be more significant. The end result would be the vehicle would move in the given direction, and nothing else around would really move at all.
An even cruder example would be to shine a bright torch out the back of your vehicle. Even though the photons have no mass, wouldn't the vehicle move forward?
 A: 
My question is, why is it impossible
  to move a mass in a given direction
  without a proportional change in the
  opposite direction?

Because otherwise the principle of conservation of linear momentum will be violated (A more accurate answer is, it is a law of nature).
The reaction less drive is a complete fraud.
A: Edward's and sb1's answers already state this well in words, but I prefer formulae:
Motion is determined by the total force $\vec F_j$ acting on a particle $j$, for which you can find a potential1 $V$ such that $\vec F_j = \vec\nabla_j V$ (with $\nabla_j$ I mean derivation with respect to the $j$th particle's coordinates). Since even in non-relativistic mechanics there is no reason to emphasize one coordinate system over the other, this means that $V$ can only depend on the positions of a particle relative to the others, i.e.
$V=V(\vec x_1 - \vec x_2, \vec x_1-\vec x_3, ... , \vec x_{N-1} - \vec x_N)$ for $N$ particles2. The result is that the sum of all forces vanishes, $\vec F = \sum\limits_j\vec F_j = \sum\limits_j\vec\nabla_j V = 0$, which for $N=2$ particles yields the famous actio = reactio. This also means that, in general, a force acting on one particle (e.g. $j=1$) is cancelled out by the total force acting on all other particles, i.e. $\sum\limits_{j\neq 1}\vec F_j = -\vec F_1$ - that is why it is impossible to move a mass in a given direction without a proportional change in the opposite direction.

1) for friction etc. generalized potentials can still be used in the Lagrangian formulation
2) I assume no external forces are present - they are caused by (maybe many) particles as well and just because you don't notice earth being attracted in your direction as well doesn't mean it doesn't happen
A: It appears to me the issue is understanding momentum conservation.

An even cruder example would be to shine a bright torch out the back of your vehicle. Even though the photons have no mass, wouldn't the vehicle move forward?

You also refer to mass in this manner in the paraphrasing of Newton's third law "proportional opposite mass/acceleration ratio in the opposite direction".  It is not the mass that matters here.  Newton's third law can be thought of as a statement of conservation of momentum.  Despite having no invariant mass, photons do have momentum.  So a light sail, or a flash light in your case, works because to conserve momentum if light is reflected off your craft or emitted from your craft, your momentum must change to compensate for the change in momentum of the light or the momentum the light carried away.
Noether's theorem is an even stronger statement, which shows that if we can describe the physics in a manner that does not depend on position (ie. you could redo this experiment 1km to the right and it wouldn't effect the results), then momentum must be conserved.  So this forbids a reactionless drive in special relativity as well.
Spatial translation symmetry becomes a bit messy in GR, since the spacetime itself is dynamic.  So your gravity wave idea could work in principle (Alcubierre drive).  However it is possible to formulate a type of energy and momentum conservation with pseudo-tensors in GR (this is how Einstein discussed it).  In this view we can keep track of the energy and momentum of gravity waves as well, and so we can still use momentum conservation for these scenarios as well.
So your question:
Why is it impossible to move a mass in a given direction without a proportional change in the opposite direction?
the answer is:  Momentum conservation forbids a reactionless drive, and momentum conservation itself follows from the translational symmetry of the physics describing a closed system.
So like free energy machines, we don't even need to see the details to know something is a misrepresentation or a scam.  We can reject such "inventions" on very general grounds.  And indeed, the U.S. patent office explicitly will not review a patent on a perpetual motion machine, or a device that could be used to build such a machine.  They may have a similar restriction on "reactionless drives", but I am not sure of that.
A: As is stated in other answers: momentum conservation forbids it. Momentum conservation must hold because it is a conserved Noether charge caused by spacetime translation continuous symmetry.
But (there is almost always a planck-sized but in any complete answer) There are situations where you can get net displacement in a curved spacetime without throwing away any propellant, or even changing the net momentum. This has been addressed in other physics.SE questions:
Can a deformable object "swim" in curved space-time?
Swimming in Spacetime - apparent conserved quantity violation
net displacement and path dependence
A: There is no such thing as a "reactionless" drive.  However, there are certain principles in physics that can be "bent" by creating reactions that have less than obvious components.  For example, the theory of unbalancing a centripetal force is sound, however all attempts that I've seen fall into two "traps".  The first is the double circle trap.  They use a device like the Dean Drive that uses a rotation and a counter-rotation to vary the radius of a centripetal force device.  Doing this creates a second centripetal force which automatically cancels out the first...these people didn't do the math.  If you calculate the forces, it cancels out.  Every time.  The second set of people create something that "bumps" an object, overcoming friction for a moment or two, then repeating.  This is in no way reactionless.  This is impact and friction, plain as day.  The solution lies in creating an object with a single centripetal force, a variable radius, and a constant angular velocity.  This presents various engineering problems, but the actual result is still NOT reactionless.  It is the action of an accelerating object, and the perceived force (centripetal force) is acting on the object according to the law of conservation.  
More recent developments include another type of reactionless drive in development by the Chinese that uses a magnetron to input microwaves into a specially shaped chamber.  Having looked at their device, I fully understand why they have achieved milliNewtons of force from kiloWatts of power, and it is because they haven't the slightest notion of why it works in the first place.  I refer you to the law of conservation again--and to reflection.  When a wave bounces off of a surface, some of its force is translated to the surface.  The angle of incidence equals the angle of reflection, and the vector of the force translated into the object splits the difference.  The chamber the Chinese are using is flat on two sides, and the sides are joined by a conical tube.  The microwaves are not being mased (lined up like lased photons), so they are bouncing around willy-nilly in the chamber, and fewer waves hit the small end than the large end.  The waves that hit the cone translate their force at an angle that is not the same as the small end, and therefore the force that directly opposes the force emanating from the large end is diffused by changing its vector.  If they mased their waves and shaped their chamber, they'd see much better results.  If I had a lab or funding to continue my research, I could show you five other ways to cheat the law of conservation without violating it.  Why this technology is still in development I'll never understand.  It's basic physics.  For some reason the scientific community throws a blanket over it and says it's not possible.  It's not impossible to create motion without expelling rocket fuel, but it needs to be looked at by someone with real knowledge instead of a community of untrained amateur engineers who don't do their math and don't understand the physics.   
A: Suppose two bicycle chains on sprockets and a frame in a J-shape were pulled inwards in a mirror image of each other? They would cancel out both linear and rotational forces and mass would be transfered linearly "round the U bend" without a linear opposite reaction(?)
The structure could then be rotated to a new position, and the process repeated.
A: 
Noether's theorem is an even stronger statement, which shows that if we can describe the  physics in a manner that does not depend on position (ie. you could redo this experiment > 1km to the right and it wouldn't effect the results), 

It is precisely Noether's theorem that could be the weak link that allows free energy: In the case of energy you need time symmetry for conservation. If your interaction is somehow time assymmmetric, then you might be able to avoid conservation of energy.
A: Actually it IS possible to have a reaction-less force: in a similar way to free energy, precisely because of Noether's theorem mentioned above. In the case of energy, Noether says if the force is not constant in time, energy need not be conserved.  In certain (electro-)magnetic configurations, this is indeed the case and several companies are working towards exploiting this effect. One of the effects also allows momentum conservation to be violated. 
