Weight‐adjusted inner products are easily invertible approximations to weighted L2 inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time‐domain method for wave propagation which is low storage, energy stable, and high‐order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight‐adjusted DG methods to the case of matrix‐valued weights, with the linear elastic wave equation as an application. We present a DG formulation of the symmetric form of the linear elastic wave equation, with upwind‐like dissipation incorporated through simple penalty fluxes. A semidiscrete convergence analysis is given, and numerical results confirm the stability and high‐order accuracy of weight‐adjusted DG for several problems in elastic wave propagation.