A novel method is presented to calculate the reflection coefficient of sound from a semi-infinite periodic system of elastic layers. Each layer may consist of sublayers, and may be either isotropic, or transversely anisotropic. The reflection coefficient is given in an explicit form for alternating solid and liquid layers at an oblique incidence of a sound wave as well as for all solid layers at normal incidence. For all solid layers at oblique incidence a quadratic matrix equation with respect to the input impedance matrix is obtained. The method of solving this equation and the coefficients of a transfer matrix for transversely isotropic layers are presented in the Appendices. Resonance properties of the system of layers follow from the results of computations of the frequency and angular dependencies.