A mathematical model is presented for the propagation of structural waves on an in"nitely long, periodically supported Timoshenko beam. The wave types that can exist on the beam are bending waves with displacements in the horizontal and vertical directions, compressional waves and torsional waves. These waves are a!ected by the periodic supports in two ways: their dispersion relation spectra show passing and stopping bands, and coupling of the di!erent wave types tends to occur. The model in this paper could represent a railway track where the beam represents the rail and an appropriately chosen support type represents the pad/sleeper/ballast system of a railway track. Hamilton's principle is used to calculate the Green function matrix of the free Timoshenko beam without supports. The supports are incorporated into the model by combining the Green function matrix with the superposition principle. Bloch's theorem is applied to describe the periodicity of the supports. This leads to polynomials with several solutions for the Bloch wave number. These solutions are obtained numerically for di!erent combinations of wave types. Two support types are examined in detail: mass supports and spring supports. More complex support types, such as mass/spring systems, can be incorporated easily into the model.