A combined wavepacket-path integral method is proposed for computing the time evolution of nuclear coordinate wavepackets on several nonradiatively coupled potential energy surfaces. In order to make the technique useful for multidimensional problems, the method uses the Gaussian wavepacket dynamics (GWD) algorithm (Heller, E. J.; J. Chem. Phys. 1975, 62, 1544) to perform nuclear coordinate wavepacket evolution on successive individual surfaces as prescribed by the path integral formalism. Since the GWD algorithm is not exact for anharmonically coupled potential energy surfaces, this introduces some error into the calculation. However, as long as the GWD algorithm is sufficiently accurate to treat single-surface propagation on all of the relevant potential energy surfaces, the proposed method is expected to yield useful results. A simple illustration is given using a one-dimensional model of a direct photodissociation process involving two nonradiatively coupled electronic states. Good agreement with numerically exact results is obtained, even in the case of strong, and position-dependent, nonradiative coupling.