We develop a theory of elastic waves in oriented monodomain nematic elastomers. The effect of soft elasticity, combined with the Leslie-Ericksen version of dissipation function, results in an unusual dispersion and anomalous anisotropy of shear acoustic waves. A characteristic time scale of nematic rotation determines the crossover frequency, below which waves of some polarizations have a very strong attenuation while others experience no dissipation at all. We study the anisotropy of low-frequency Poynting vectors and wave fronts, and discuss a "squeeze" effect of energy transfer nonparallel to the wave vector. Based on these theoretical results, an application, the acoustic polarizer, is proposed.