If gravity is a wave, what medium does it pass through? Is there a medium that it can not pass through? I listened to a podcast discussing gravity and had this question. My apologies if it has been aptly answered somewhere else.
 A: Gravity is described classically by General Relativity, and yes it also has gravitational waves :

Gravitational waves are disturbances in the curvature of spacetime, generated by accelerated masses, that propagate as waves outward from their source at the speed of light. They were proposed by Henri Poincaré in 1905 and subsequently predicted in 1916 by Albert Einstein on the basis of his general theory of relativity. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation.

So it is a wave distortion of space time due to changes in gravitation, not a medium in the sense of matter.
In efforts of quantizing gravity the graviton is proposed to build up the classical general relativity wave in a way similar to photons building the electromagnetic wave.
A: In general relativity gravitation is described by the curvature of spacetime and does not need any medium. 
Objects bend the spacetime and therefore other objects feel their gravitational force. 
