We show that the resonant coupling of phonons and magnons can be exploited to generate spin currents at room temperature. Surface acoustic wave pulses with a frequency of 1.55 GHz and duration of 300 ns provide coherent elastic waves in a ferromagnetic thin-film-normal-metal (Co/Pt) bilayer. We use the inverse spin Hall voltage in the Pt as a measure for the spin current and record its evolution as a function of time and external magnetic field magnitude and orientation. Our experiments show that a spin current is generated in the exclusive presence of a resonant elastic excitation. This establishes acoustic spin pumping as a resonant analogue to the spin Seebeck effect.