Practical obstacles, mainly.
First, it would be *really* hard to vibrate a mass large enough to produce any non-trivial amount of gravity.
Second, although it seems a strong force, gravity is actually *incredibly* weak compared to electromagnetism, and very, very hard to detect if you're talking about masses less than the size of a moderately large mountain.
Incidentally, this is exactly why researchers have yet to be able to devise an experiment detecting gravity waves, too. If you *could* detect gravity waves, you could make a communication device using them. But to date no one's yet been able to. It's like trying to pick up the sound of a crash-landing mosquito across the Pacific ocean.
Then, how would one modulate the baseband? Or would we have to use CW?