We study theoretically the geometrical and temporal commensurability oscillations induced in the resistivity of two-dimensional electrons in a perpendicular magnetic field by surface acoustic waves (SAWs). We show that there is a positive anisotropic dynamical classical contribution and an isotropic nonequilibrium quantum contribution to the resistivity. We describe how the commensurability oscillations modulate the resonances in the SAW-induced resistivity at multiples of the cyclotron frequency. We study the effects of both short-range and long-range disorder on the resistivity corrections for both the classical and quantum nonequilibrium cases. We predict that the quantum correction will give rise to zero-resistance states with associated geometrical commensurability oscillations at large SAW amplitude for sufficiently large inelastic scattering times. These zero resistance states are qualitatively similar to those observed under microwave illumination, and their nature depends crucially on whether the disorder is short or long range. Finally, we discuss the implications of our results for current and future experiments on two-dimensional electron gases.