This talk will introduce a recently developed spectral numerical algorithm for the solution of Cauchy-Navier elasticity equations on general three-dimensional domains. Based on a recently introduced “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions, the proposed high-order approach can yield solutions within a prescribed error tolerance by means of significantly smaller discretizations and computing times than those required by other methods. Additionally, the new methodology entails mild CFL constraints; runs at a cost that scales linearly with the discretization sizes; and lends itself easily to efficient parallelization in distributed-memory computing clusters. Results applying the new algorithm to problems of isotropic elastodynamics in the context of wave scattering in thin plates for non-destructive evaluation of materials as well as problems in seismic wave propagation on three-dimensional topographies will be presented. Ongoing and future extensions into unbounded domains, hybrid implicit-explicit temporal discretizations, layered solid media, and complex material interface geometries will additionally be discussed.