Free vibrations and plane waves are investigated in a linear approximation in a thin elastic anisotropic infinite plate of constant thickness. A general anisotropy described by 21 elastic moduli is considered. It is assumed that the moduli and the density do not depend on the tangential co-ordinates, and they may depend on the thickness co-ordinate. Multi-layered and functionally graded plates are not excluded from consideration. An asymptotic expansion in power series of a small thickness parameter, μ, of a harmonic in tangential directions solution of 3D equations of the theory of elasticity is built in assumption that the length of wave essentially exceeds the plate thickness. For the fixed values of the wave numbers there exist only three long-wave solutions: one bending low-frequency solution, and two tangential solutions. The dispersion equations are built with the second order accuracy in μ. For the bending solutions the strong dependence of the frequency on the length of wave is typical, and the tangential waves propagate with the small dispersion. Partial cases of anisotropy are considered.