We study epitaxial growth onto a nonplanar substrate by use of phenomenological equations of motion. The substrate is chosen as a periodic sequence of mesa structures composed of well-defined crystallographic facets. Although surface diffusion is not treated explicitly, the problem is formulated at a level of sophistication sufficient to account for the atomistic kinetic processes of deposition, desorption, adatom exchange with kink sites, and interfacet migration. Numerical integration of the final kinetic equations reveals a variety of growth morphologies depending upon the relative rates of these processes. The predictions of our analysis should be amenable to direct experimental test.