This paper suggests a mathematical model for decaying vibrations of layered plates formed by a finite number of arbitrarily oriented orthotropic viscoelastic layers of polymer composites arranged into an anisotropic structure with a layer of “stiff” isotropic viscoelastic material applied on one of its outer surfaces. The model is based on Hamilton variation principle, first-order shear deformation laminated plate theory (FSDT) and the viscoelastic correspondence principle of the linear viscoelasticity theory. In the description of the physical relationships between the materials of the layers forming orthotropic polymeric composites, the effect of vibration frequency and ambient temperature is assumed as negligible, whereas for the viscoelastic polymer layer, temperature-frequency relationship of elastic dissipation and stiffness properties is considered by means of the experimentally determined generalized curves. As a particular case of the general problem, neglect of medium surface straining in direction of one of the axes of the plate yielded the movement equations for Timoshenko beam with a layer of “stiff” isotropic viscoelastic polymer on one of its outer surfaces. Mitigation of Hamilton functional makes it possible to describe decaying vibration of anisotropic structures by an algebraic problem of complex eigenvalues. The system of algebraic equation is generated through Ritz method using Legendre polynomials as coordinate functions. First, real solutions are found. To find complex natural frequencies of the system, the obtained real natural frequencies are taken as input values, and then, by means of the 3rd order iteration method, complex natural frequencies are calculated.