Abstract: Free vibrations and plane waves are analyzed in the linear approximation for a thin elastic, anisotropic infinite plate having constant thickness. A general anisotropy described by 21 elastic moduli is considered. It is assumed that the moduli of elasticity and density are independent in the tangential coordinates but can depend on the coordinate along the thickness of the plate. Multilayer and functionally gradient plates are also considered. Assuming that the wavelength significantly exceeds the thickness of the plate, an asymptotic power expansion is obtained for a small thickness parameter for a harmonic solution of the system using the three-dimensional equations in tangential coordinates provided by the elasticity theory. For fixed values of wave numbers, there are only three long-wave solutions available: one low-frequency bending and two shearing ones. Dispersion equations are obtained for these solutions with accuracy to the terms of the second order of smallness in the dimensionless thickness. Bending solutions are characterized by a strong dependence of frequency on the wavelength, while the tangential waves propagate with low dispersion. Particular types of anisotropy are considered.