The paper describes a theoretical study on the speed of torsional elastic waves propagating in a circular rod whose material properties vary periodically as harmonic functions of the axial coordinate. An approximate solution for the phase speed has been obtained by using the perturbation technique for sinusoidal modulation of small amplitude. This solution shows that the wave speed in the nonuniform rod is dependent on the wave frequency as well as the periodic variation of the material properties. It implies that the torsional waves considered in this paper are dispersive even in the fundamental mode.