Weight-adjusted inner products are easily invertible approximations to weighted L2 inner products and mass matrices. These approximations make it possible to formulate very simple time-domain discontinuous Galerkin (DG) discretizations for wave propagation based on the the energy of the system. The resulting methods are low storage, energy stable, and high-order accurate for acoustic and elastic wave propagation in arbitrary heterogeneous media and curvilinear meshes. We conclude with numerical results confirming the stability and high-order accuracy of weight-adjusted DG for acoustic, elastic, and coupled acoustic-elastic waves.