We have developed a method for modeling transient electromagnetic data acquired on land or at sea. The 3D 3C time response of the electric wavefield is simulated by considering the spatial and temporal responses separately and using numerical methods best suited for each. We have used the rapid expansion method to develop the 3C electric wavefield time response from the spatial responses found using a pseudospectral method. The results are free of numerical dispersion and accurate to the Nyquist frequency in time and space. The resulting diffusive field is a weighted sum of Chebyshev polynomials that exhibit a wave-like character and are very sensitive to small perturbations in the medium. The method is intrinsically parallel, which leads to computational efficiency. Numerical results are compared favorably with the analytic response and 1D methods even though the computations are innately 3D.