A molecular dynamics simulation (MD) of a generic model for polymers adsorbed on a discrete surface is presented. We investigate the effect of surface roughness on the film's diffusion constant, its influence on aging, and other properties. The model reproduces many experimentally observed features such as logarithmic aging of structural properties with time, a power law aging of the diffusion constant D, and the anomalous dependence of mobility with surface coverage. It is shown that the glassy dynamics is intimately linked to the roughness, which can be controlled by changing the interaction potential between surface atoms and adsorbed particles. We observe a logarithmic increase in the structure factor with aging time and a power law decay in D only if the corrugation is sufficiently high. For a smooth surface, no such time dependence is observed, even if adhesion and thus confinement is increased. We also address the question as to why the mobility of adsorbed polymers can initially increase with surface coverage Gamma when Gamma is low.