We report on the first systematic study of phonon propagation in nanostructured composite polymer multilayer films as a function of periodicity and composition using Brillouin light scattering and numerical simulations. The high sensitivity of phonon dispersion to structure and composition allows the probing of the mechanical properties down to the single-layer level. We observe a strikingly different dependence of the longitudinal and shear moduli on confinement effects in the polymer nanolayers. In addition, temperature dependent measurements of sound velocities reveal the presence of distinct glass transition temperatures, indicative of phonon localization in films with large layer thicknesses in agreement with theoretical predictions.