  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
($\tau_{D}^{-1}$). However, $\tau_{D}^{-1}$ 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 $\ell_{ph}$ smaller than a critical value, imperfection
scattering dominates and the drag rate varies as $ln (\ell_{ph}/d)$ over many
orders of magnitude of the layer separation $d$. When $\ell_{ph}$ 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 $d$-dependence of the drag rate.
