We report on Brillouin measurements of the phonon dispersion relation of a three-dimensional hypersonic phononic crystal. The phononic crystal, comprised of close-packed polystyrene spheres embedded in a background matrix of solid polydimethylsiloxane, was fabricated using a self-assembly technique. The experimentally measured dispersion relation reveals the band folding effect and the formation of a band gap for longitudinal waves at the boundary of the Brillouin zone. Theoretical calculations, based on the finite element method and the finite-difference time-domain method, are employed to interpret experimentally observed phonon modes and provide insight on important features of the solid/solid hypersonic crystals.