There in fact is a large project underway to determine precisely this, as a test of Einstein's theory of relativity. It is called Gravity Probe B, and is an earth-orbiting satellite designed to remain in polar orbit over the Pacific Northwest for two years or so. It will use four extremely precise gyroscopes to measure space-time and changes in it, and whether the Earth 'drags' changes in space-time around with it as it orbits. One of the prime questions it aims to answer is whether or not gravity propagates instantaneously or at the speed of light.

It is possible for gravity to propagate instantaneously without violating any of the laws of relativity (or other physical laws) as long as it isn't possible to transmit information using this phenomenon. This would likely mean that gravity is not a wave or particle; if it were, then it would be subject to manipulation that could be used to transmit information faster than light. If, however, gravity is a 'bending' of space-time, then it would require other violations of 'law' to use it to transmit information - for example, in order to have a gravitic signal one would need to be able to create energy and/or matter, or to destroy the same such that it suddenly stopped bending space by its presence. Since energy cannot be created or destroyed, it would not be possible to send information using this effect.

http://einstein.stanford.edu/index.html

...it is a NASA mission planned for launch from Vandenberg AFB on a Delta-II rocket, sometime in fall 2002.

The following in response to a vanished writeup:

Energy does, in fact, 'have' gravity. Matter is condensed energy, as exemplified by E = mc2. Enough energy in one place bends space, just like matter. Although you can change matter into energy, absolutely right, that doesn't change anything! The 'center of mass' of this wave of matter and energy doesn't move faster than light. In a perfectly spherical explosion, it doesn't move at all. Ergo, such a conversion wouldn't tell us anything about gravity's travels.

We don't, in fact, know if gravitons exist. If they do, then yup, we have a problem with gravity moving FTL. However, if it isn't composed of gravitons but is rather a field distortion, then sure it could.

If a distortion, then (as we saw above) we can't transmit information FTL, since the only way to do that would be to create or destroy energy/matter so as to cause 'pulses' in the field. Since we're pretty sure we can't do that, we're still in the clear. If we could in fact drop mass into, say, hyperspace, so that it no longer affected our spacetime, then that would be the perfect experiment on the subject of gravity's speed.

Note: Okay, tdent has backed me to a wall. If gravity is FTL, you should be able to transmit information by semaphore, or by wagging a mass back and forth. tdent points out that in such a case, the vector of its gravitational field should change instantly, so that while it might require really really sensitive instrumentation to detect at long distances, there's no theoretical reason the change couldn't transmit information. Hm. Back to the metaphor drawing board.