In-plane phonon propagation in a 1D periodic polymeric film comprised of poly(methyl methacrylate) and poly(ethylene terephthalate) layers was studied using Brillouin light scattering. In contrast to homogeneous-medium-like behavior, five acoustic-like modes with constant phase velocities plus an additional high-frequency localized mode were observed. Finite element analysis was used to compute the theoretical phonon dispersion relation and provide interpretation of the observed propagation modes. Complex features of the phonon dispersion relation were associated with elastic waves propagating within the individual periodic polymer layers. The temperature dependence of sound velocities of these modes was then used to estimate glass transition temperatures of individual nanoscale polymer layers in a sandwiched multilayer assembly.