On the basis of Hamilton variation principle, Bolotin multilayer plate theory and the elastic-viscoelastic correspondence principle in the linear theory of viscoelasticity the mathematical description of damped vibrations of laminated polymer quasihomogeneous plates is suggested. The elastic eigenvalue problem is solved on the basis of the Ritz's method. An iteration method of the third precision order is used to find the frequency equation complex roots for a damped vibration problem. The convergence estimation of the elaborated mathematical description is shown. The influence of orientation of rigid layer reinforced fibers on numerical values of natural frequencies and loss factors is discussed.