The response of a circular viscoelastic plate to a transient temperature field varying along the radius and through the thickness is investigated. The plate, initially stress free at an elevated temperature, is suddenly placed in an environment with Newtonian cooling on all surfaces. The transient and residual stresses and displacements are analyzed. The material is assumed linear and thermorheologically simple, with the relaxation response approximated by Dirichlet series; inertia and thermomechanical-coupling effects are neglected. This leads to an efficient solution algorithm that eliminates the history storage requirements. The resulting coupled integrodifferential equations are solved at each time step by a superposition method. Numerical results for float glass, polymethyl methacrylate, and polyisobutylene are given.