We have determined by first principles the athermal elastic constant tensor, single crystal and aggregated acoustic velocities, and anisotropy of the recently discovered post-perovskite MgSiO3 polymorph. This phase may be the most abundant mineral in the D″ region. We find that the post-perovskite phase has similar bulk modulus and larger shear modulus than perovskite at relevant pressures. This phase is remarkably anisotropic. Analyses of horizontal shear wave splittings in transversely isotropic aggregates indicate that diverse lattice preferred orientations could produce large variation in anisotropy consistent with seismic observations in D″.