An explicit integration algorithm for computations of discontinuous wave propagation in two-dimensional and three-dimensional solids is presented, which is designed to trace extensional and shear waves in accordance with their respective propagation speeds. This has been possible by an orthogonal decomposition of the total displacement into extensional and shear components, leading to two decoupled equations: one for the extensional waves and the other for shear waves. The two decoupled wave equations are integrated with their CFL time step sizes and then reconciled to a common step size by employing a previously developed front-shock oscillation algorithm that is proven to be effective in mitigating spurious oscillations. Numerical experiments have demonstrated that the proposed algorithm for two-dimensional and three-dimensional wave propagation problems traces the stress wave fronts with high-fidelity compared with existing conventional algorithms. Copyright © 2013 John Wiley & Sons, Ltd.