This paper is concerned with some aspects of the dynamic response of a three‐dimensional linear viscoelastic medium to prescribed boundary conditions. The constitutive equations, expressed in operator form, are applied to a three‐dimensional continuum and are combined with the equations of motion to yield a system of linear partial differential equations. This system may be expressed in terms of a scalar and vector potential and parameters called “generalized velocitiesc1*andc2*” (operators describing the properties of the medium). The free radial vibrations of a viscoelastic sphere and spherical shell are worked out for the prescribed boundary conditions. In particular the phase velocity is obtained as a function of the wave number and a complex frequency. The wave number is obtained from the roots of a transcendental equation. The free longitudinal vibrations of a viscoelastic cylinder are determined by solving the equations for the scalar and vector potentials with the appropriate boundary conditions. The phase velocity is obtained for the Voigt and Maxwell models. In addition, asymptotic results are obtained which reduce to either the thin rod or the Pochhammer approximation.