Would it be possible to transmit information through gravitational waves?

First it's important to note that gravitational waves do require energy to produce. A good example of this is a binary pulsar, where the emission of gravitational waves carries energy away so the two pulsars spiral in towards each other and will eventually merge.

Having said this, it is theoretically possible to modulate a gravitational wave and use it to transmit information. You don't need a mass modulator, you just need something with a changing quadropole moment - the simplest example of this is a spinning dumbbell, and indeed this is basically what the binary pulsar system is. If you can change the rotation frequency you can frequency modulate the gravitational wave.

However gravitational waves are exceedingly hard to generate in the sense that very little of the energy of your system is carried away as gravitational waves. It doesn't seem likely we'll ever use gravitational waves for transmitting information.

The JASONs put together a reasonably extensive report on the use of high frequency gravitational waves for anything practical (including communication) in 2008. In the abstract they state that

the following are infeasible in the foreseeable future: detection of the natural “relic” HFGW, which are reliably predicted to exist; or detection of artificial sources of HFGW. No foreign threat in HFGW is credible, including: Communication by means of HFGW; Object detection or imaging (by HFGW radar or tomography); Vehicle propulsion by HFGW; or any other practical use of HFGW.

Here HFGW stands for high frequency gravitational waves, which includes the frequencies necessary for useful communication.