The effect of a patterned surface on the phase separation kinetics of a thin polymer film has been investigated using the Cahn-Hilliard-Cook model in three dimensions with the addition of a short range surface potential. We have observed pattern-induced spinodal waves perpendicular to the surface creating "checkerboard"-like composition fluctuations for a wide range of patterns used in this study. The number of such layers strongly depends on the magnitude of the thermal noise. For sufficiently thin films, where the film thickness is smaller than the spinodal wavelength, spinodal decomposition can be arrested if the surface potential and the characteristic size of the pattern are chosen accordingly, enabling the transfer of surface patterns to the film material. The kinetic pathways through which the equilibrium states are reached delicately depend on the particular pattern, its size, and the film thickness.