Abstract: A mathematical model of damped vibrations of three-layer plates formed by two rigid anisotropic layers and a soft middle isotropic viscoelastic polymer layer is proposed in this paper. The model is based on the Hamilton’s variational principle, the refined theory of first-order plates, the model of complex modules, and the principle of elastic-viscoelastic correspondence in the linear theory of viscoelasticity. The frequency-temperature dependence of the elastic-dissipative characteristics is considered negligible for rigid layer materials; however, this dependence is taken into account for the viscoelastic polymer of the soft layer. By minimizing the Hamilton functional we reduce the problem of damped vibrations of anisotropic structures to the algebraic problem of complex eigenvalues. The Rietz method using the Legendre polynomials as coordinate functions is applied to form the system of algebraic equations. The real solutions are found. When determining the complex natural frequencies of the plate, real natural frequencies obtained are used as their initial values, and then the complex frequencies are calculated by the third-order iteration method. The results of the study of the convergence of the numerical solution are discussed. The estimation of reliability of the mathematical model and the numerical solution method obtained by comparing the calculated and the experimental values of natural frequencies and loss factors is presented.