Gravitational wave is produced by change in gravitational field, source. If something is moving away from me at constant speed, its gravitational field will vary. But why only accelerating bodies produce GW?
Because radiation from such a body undergoing inertial motion would gainsay Galileo's postulate of relativity, i.e. that there is no measurement one can make within an inertial frame that could detect that frame's motion relative to another inertial frame. Gravitational radiation would bear energy away from the body, and in principle (at least from the classical paradigm that holds in special and general relativity) this energy loss could be measured from within the frame.
If you think about it, the proposition that bodies emit radiation when in uniform motion relative to your own inertial frame is a tacit assumption that your own frame is privileged.
If you are looking for an answer with more "technology" applied, then, as user Danu comments, the lack of radiation from uniformly moving bodies is not special to gravitation: there is an approximation to GR called Gravitoelectromagnetism which is almost exactly like Maxwellian electrodynamics and can be derived from assuming a delay, propagating at speed $c$, in gravitational effects. It's kind of like Coulomb's electrostatic law modified to heed the locality requirements of special relativity. In this picture, the gravitational field lines of a moving body are distorted in a relatively uniformly moving frame, but their shape does not change and the whole solution (essential like that described the Liénard-Weichert Potential for zero acceleration) is a field of constant shape translating uniformly with the body. One can easily see from this picture that only accelerating bodies can radiate energy: a field of constant shape whose only change is uniform, rigid translation with the body producing it must have a constant energy and there can be no radiation.
One should be warned, however, that the gravitoelectromagentic approach grossly overestimates gravitational radiation power radiated from accelerating bodies by using the analogue of Larmor's formula / Liénard-Weichert Potential. Such an approach gives several gigawatts radiation from the Earth, whereas full GR calculates around about 200W. (Incidentally, the several gigawatt figure is also too small to measure for something Earth-sized).
Gravitational wave is a concept predicted by General Relativity by Albert Einstein. The very first point to look is that how a theory extended from explaining the symmetry of physical laws in inertial frames to include accelerated frames become a theory of gravitation. This is the starting point of General Relativity. It's known as equivalence principle. It says that gravity is an acceleration effect that determines the geometry of spacetime. You cannot distinguish, if put in a sealed container, that if you are in some gravitational field or whether the container is accelerating upwards. You cannot distinguish your inertial mass and gravitational mass.
So, with the aid of this, Einstein said that what massive substances can do on spacetime, the accelerating bodies can also do. Gravitational waves are the gravitational potential energy radiated by a changing gravitational field source. As we say that electromagnetic fields can be produced by charges moving at uniform velocity, electromagnetic radiation requires the charge to accelerate, same is for gravitational waves. A gravitational field can be created by mass and energy distribution. But, to cause ripples in spacetime, the field should be fluctuated rapidly (think of the ripples you can make on water). As a simple case, the massive object should vibrate. This means, the body is accelerating.
So to see any effect of gravity, you must see an acceleration. Then what's so special about GW?
I think the other answers are sufficient, but I would like to leave my two cents just to clarify a few possible concerns.
I can see you imagining that moving objects in your reference frame leave a gravitational "wake", like a boat moving in water, but gravity and relativity don't quite work this way.
To grossly simplify, from the moving object's perspective, you are the one moving and it staying still. However, from your perspective, your gravity is equal in all directions, and from the object's perspective, so is its gravity.
Now, gravitational waves come from acceleration. If there is no acceleration, there are no gravitational waves.
"But when the object passes me, doesn't its gravitational 'wake' cause me to accelerate?" If you have mass, the answer is yes, but because you have mass, your gravity also affects the object and causes it to accelerate. In this case, yes gravitational waves are generated, but only because you and the object interacted and caused acceleration.
If you were, however, a massless/energy-less/basically-non-existant observer, then the gravity would do nothing to you, and you would have no affect on the object. There would be no acceleration, and therefore, no gravitational waves would be generated.
I hope that makes sense.