Canonical variational transition state theory and a multidimensional semiclassical tunneling method were used to calculate the surface diffusion rate constants of H, D, and T on the (100) surface of a model metal, nominally Cu. In the present study, we especially examined the influence of metal motions on the kinetic isotope effect for this process. We have employed the embedded cluster approach with a cluster size of 28 metal atoms. The adsorbate-substrate and substrate-substrate interactions are modeled by pairwise potential functions. The results show that including the metal zero-point motions has a negligible effect on the kinetic isotope effect predicted by the rigid-lattice model, but the inclusion of nonzero metal vibrational amplitudes in the semiclassical tunneling path has a large effect. We call this phonon-assisted tunneling. In the low-temperature region, where phonon-assisted tunneling is most important, the isotope effects were found to be smaller than those predicted by the rigid-lattice model, whereas at higher temperatures, they are found to be larger than the lattice predictions. This occurs despite the fact that the phonon-adsorbate interactions decrease the effective reduced masses of the adsorbates in the tunneling region.