The surface morphology of homoepitaxial GaAs layers grown by molecular beam epitaxy on random and periodically textured substrates has been measured by atomic force microscopy and elastic light scattering. The random texture was obtained by thermal evaporation of the surface oxide and the periodic texture consisted of one-dimensional grating patterns fabricated by holographic lithography. The time evolution of the surface morphology was simulated numerically with a nonlinear growth equation that includes deposition noise and anisotropy in the surface diffusion. The surface of the random substrate develops shallow mounds as the large amplitude initial texture smooths out, an effect that has previously been attributed to unstable growth.