A mathematical model of damped oscillations of three-layerplates formed by two rigidanisotropic layers and a soft middle isotropic layer of a viscoelastic polymer is proposed.Each hard layer is an anisotropic structure formed by a finitenumber of randomly ori-ented orthotropic viscoelastic composites layers. The model is based on the use of theHamiltonian variational principle, the refined theory of first-order plates (Reissner-Mindlintheory), the model of complex modules and the principle of elastic-viscoelastic correspon-dence in the linear theory of viscoelasticity. When describing the physical relationships ofhard layer materials, the influence of the vibration frequency and the ambient temperatureis considered negligible, while for the soft layer of a viscoelastic polymer, the temperature-frequency dependence of the elastic-dissipative characteristics is taken into account basedon experimentally determined generalized curves. As a special case of the general problem,by neglecting the deformation of the middle surfaces of the rigid layers in one of the di-rections of the axes of the rigid layers of a three-layer plate, the equations of longitudinaland transverse damped oscillations of a globally orthotropic three-layer beam are obtained.Minimization of the Hamilton functional allows us to reducethe problem of damped vibra-tions of anisotropic structures to the algebraic problem ofcomplex eigenvalues/