We use the Kubo formalism to evaluate the contribution of acoustic-phonon exchange to the frictional drag between nearby two-dimensional electron systems. In the case of free phonons, we find a divergent drag rate (Formula presented) However, (Formula presented) becomes finite when phonon scattering from either lattice imperfections or electronic excitations is accounted for. In the case of GaAs quantum wells, we find that for a phonon mean free path (Formula presented) smaller than a critical value, imperfection scattering dominates and the drag rate varies as (Formula presented) over many orders of magnitude of the layer separation (Formula presented) When (Formula presented) exceeds the critical value, the drag rate is dominated by coupling through an electron-phonon collective mode localized in the vicinity of the electron layers. We argue that the coupled electron-phonon mode may be observable for realistic parameters. Our theory is in good agreement with experimental results for the temperature, density, and (Formula presented) dependence of the drag rate.