Do accelerating masses generate gravitational waves? The accelerating charge can radiate electromagnetic waves, then, can the accelerating masses radiate gravitational waves?
 A: Yes, this is what happens, but electromagnetic radiation can be generated if there is a time-dependent dipole moment of the charge. In GR, a time-dependent dipole moment does not lead to any radiation, and the leading order contribution is the quadrupole. So a mass oscillating back and forth on a line will tend not to generate much radiation, compared to a mass moving in a circle (or two point masses, as in a binary).
A: Yes. The prime example of generating gravitational waves are two bodies orbiting around each other, for example the Hulse-Taylor binary, which was the first indirect discovery of gravitational waves in 1974 and lead to a Nobel Prize in 1993. (Keep in mind, that even if a body orbits on a perfect circle, there is acceleration as the velocity vector changes even though its magnitude does not.)
As Andrew already wrote, there is no dipole radiation of gravity, a main difference to electromagnetic radiation. This is due to the fact that there is no negative mass, and so every dipole vanishes when changing to the center-of-mass system. The quadrupole moment doesn't have to vanish though. For example, the Earth has a nonvanishing one due to not being a perfect sphere and being flattened a bit.
Finally, acceleration does not always cause gravitational waves. For example, a spherically symmetric (like a pulsating star) or rotationally symmetric acceleration (like a rotating disc) won't. The first is a consequence of the Birkhoff theorem.
