The phenomenon of dynamic polarization echoes has been studied in powders of several piezoelectric materials at frequencies in the VHF band and in the C, X, and Ku microwave bands. The phenomenon is found to be a consequence of the anharmonicity associated with the mechanical oscillations of the individual particles of the powder. The phenomenon does not result from a parametric coupling of the applied rf fields to the mechanical-oscillator modes as might be expected by analogy to spin echoes and other types of polarization echoes. The anharmonic-oscillator model including damping has been developed in the small-signal limit for finite pulse widths. Although different sources of the anharmonicity cannot be distinguished in this limit, the experimental data unambiguously distinguish between the anharmonic-oscillator model and the parametric field-mode interaction. Under most experimental conditions the small-signal limit is not valid and the echo shapes, dependence on pulse amplitudes, and decay properties exhibit complex behavior. The quantitative understanding of this behavior awaits further detailed calculations. Multiple two-pulse echoes are detected at high rf powers. In the anharmonic-oscillator model multiple echoes arise naturally and the predicted decay behavior is in agreement with experiment. However the decay time is found to be strongly influenced by damage on the surface of the individual particles and by gases or liquids which surround the particles. Measurements of T2-1 in a variety of gases with different acoustical impedances are in agreement with a simple calculation based upon the acoustical impedance mismatch between the solid and the gas. Measurements of T2-1 versus frequency substantiate this model and indicate that the decay mechanism is internal to the particles when immersed in a high vacuum. Measurements in diluted samples demonstrate that interparticle interactions play no role in the echo formation process or in the observed echo behavior. © 1978 The American Physical Society.