Major reshaping of electromagnetic MM-wave beams transmitted at resonant frequencies through high-quality multilayer structures is demonstrated. Beam reshaping emerges due to efficient excitation of intrinsic quasi-optical modes with complex spatial profiles by an incident beam. This leads to a significant increase of width and distortion of shape of transmitted beams, which grow with increasing likelihood of excitation of higher-order modes. The beam shape is extremely sensitive to imperfections of real structures and can become complex and asymmetric even at the normal-angle transmission of symmetric incident beams. The effect is of importance for the resonant MM-wave and THz spectroscopy of low-loss dielectric materials.