We present a spectral numerical algorithm for the fast solution of elastodynamics problems in general 3D domains based on a FFT-speed Four-ier Continuation (FC) approximation for accurate Fourier expansion of non-periodic functions. The high-order methodology yields physically correct solutions including those with traction conditions on curved boundaries. The approach is essentially without dispersion errors; entails mild CFL constraints; runs at a cost scaling linearly with discretization size; and can be efficiently parallelized for computing clusters.