We have used Brillouin light scattering (BLS) to investigate the mechanical properties of thin polystyrene (PS) and polyisoprene (PI) films incorporated in a multilayer geometry consisting of alternating layers of the two polymers. All samples had a total thickness h similar to 1600 Angstrom, and consisted of between 2 and 10 individual polymer films; individual layer thicknesses were as small as 160 Angstrom. Ellipsometry was used to determine the thickness of the PS and PI layers. The velocities of several high-frequency film-guided mechanical waves were measured using BLS and the results are compared with values calculated using an effective medium approach. The effective elastic constants of the multilayered films were obtained from those determined for thick films of PS and PI. Remarkable agreement was obtained between the measured and calculated velocities even for samples in which the individual layer thicknesses were much less than the unperturbed size of the polymer molecules. These results suggest that the mechanical properties of polymers change very little even when the molecules are forced into highly confined geometries.