The propagation of sound waves in layered conductors with quasi-two-dimensional electron energy spectra is studied theoretically. It is shown that in a magnetic field H the rate of sound attenuation G oscillates with a great amplitude as a function of and of the angle between H and the sound wavevector k, if the charge carriers are capable of drifting along k, and the radius of curvature of their trajectories r is much less than the electron mean free path l but significantly exceeds the wavelength of sound . The diameter of the Fermi surface in the direction orthogonal to the vectors k and H can be determined to a high degree of accuracy from the measured period of the oscillations.