We investigate the many-body properties of graphene on top of a piezoelectric substrate, focusing on the interaction between graphene electrons and piezoelectric acoustic phonons. We calculate the electron and phonon self-energies as well as the electron mobility limited by the substrate phonons. We emphasize the importance of proper screening of the electron-phonon vertex, and we discuss the various limiting behaviors as a function of electron energy, temperature, and doping level. The effect of piezoelectric acoustic phonons on graphene electrons is compared with that of intrinsic deformation acoustic phonons. Substrate phonons tend to dominate over intrinsic ones for low doping levels at high and low temperatures.