Wave propagation in a periodically layered medium is studied in which each period consists of two layers of homogeneous anisotropic elastic materials. The layered medium occupies the half-space x $\geqslant$ 0 in which the x-axis is normal to the layers. Transient waves in the layered medium are generated by a unit step load in time applied at x = 0. A general solution that applies to any x is obtained in the form of a Laplace transform. Asymptotic solutions valid for large x are then deduced. If the applied load at x = 0 is in the direction of one of the polarization vectors for the layered medium determined here, the stress components propagate uncoupled asymptotically. For general loadings, there are three `heads of the pulses', each of which is in the form of an Airy integral.