We study theoretically the resonant transmission of acoustic phonons in a double-barrier system for phonons consisting of a bulk material sandwiched between periodic superlattices. The phonon transmission rate and the resonance condition in this system are derived analytically based on the transfer-matrix method. We show that the phonons in a frequency gap of the superlattices can be transmitted through the whole system without attenuation, if the frequency of incident phonons satisfies the resonance condition. Based on the Green’s-function method, we also show that the resonant frequencies coincide with the impurity levels associated with the lattice vibrations localized near the embedded bulk layer.