Abstract <p>The paper is devoted to the development of one of the methods for constructing solutions to problems of nonstationary waves in inhomogeneous viscoelastic bodies. A piecewise homogeneous structure with continuity conditions at the contact of components is accepted as the main type of inhomogeneity. The continuous heterogeneity of viscoelastic functionally graded materials is approximated by a layered homogeneous medium. The hereditary properties of the components are characterized by linear Boltzmann–Volterra relations with kernels of various types. The integral Laplace transform in time and the operation of its reversal are used. New forms of solutions for unsteady viscoelasticity problems for piecewise homogeneous bodies, convenient for numerical implementation, are obtained. The proposed approach is demonstrated on a dynamic problem for an elastic hollow sphere with a coating made of viscoelastic functionally graded material.</p>