The propagation of roughness from a patterned silicon surface by polymer thin films was measured as a function of film thickness, time, and surface interaction using atomic force microscopy and synchrotron X-ray reflection. In the presence of an interacting surface, the decay length of the surface modulation was much longer than that observed in simple liquids. By measuring the time dependence of the surface corrugation amplitude, we were able to extract a surface diffusion coefficient by applying a modified version of the Mullins theory for surface diffusion in crystalline solids. The measured diffusion coefficients were an order of magnitude smaller than in the bulk, and scaled as 1/M-3/2, in agreement with previous SIMS results. The results are interpreted in terms of surface interactions confining polymer chains over distances larger than the radii of gyration.