The second-order elastic wave propagation equations are solved using the spectral moments method. This numerical method, previously developed in condensed matter physics, allows the computation of Green's functions for very large systems. The elastic wave equations are transformed in the Fourier domain for time derivatives, and the partial derivatives in space are computed by second-order finite differencing. The dynamic matrix of the discretized system is built from the medium parameters and the boundary conditions. The Green's function, calculated for a given source-receiver couple, is developed as a continued fraction whose coefficients are related to the moments and calculated from the dynamic matrix. The continued fraction coefficients and the moments are computed using a very simple algorithm. We show that the precise estimation of the waveform for the successive waves arriving at the receiver depends on the number of moments used. For long recording times, more moments are needed for an accurate solution. Efficiency and accuracy of the method is illustrated by modeling wave propagation in 1-D acoustic and 2-D elastic media and by comparing the results obtained by the spectral moments method to analytical solutions and classical finite-difference methods.