Large rotations are ubiquitous in solid mechanics. It has been common practice to solve problems involving such rotations using nonlinear finite element methods, even when deformations are small. To reduce computational cost, we propose a linearized corotational finite element method which is applicable to problems in which the deformable bodies undergo only infinitesimal deformations. This framework is a linear numerical method that approximately conserves momentum and energy—even when the incremental rotation per time step is large—for elastodynamic problems with finite rotations. Moreover, the proposed method is also fully invariant under superposed rigid-body motions, second-order accurate, simple to implement, computationally efficient, and equally well applicable to particle systems, body-fitted meshes, and unfitted meshes. The fitted-mesh formulation is naturally immune to mesh distortion issues encountered in typical Lagrangian meshes. The unfitted-mesh formulation, which eliminates the need for nodes to convect with material points, employs the cut-cell method on a fixed, uniform, axis-aligned Cartesian grid as the Eulerian mesh. To our knowledge, this is the first unfitted-mesh method for elastodynamics, offering benefits for interface problems with moving boundaries, such as fluid–structure interactions. We demonstrate the effectiveness of our novel approach through a motivating example of an elastic pendulum and two numerical examples on elastic continua. The results demonstrate conservation of linear and angular momentum, conservation of total energy, and second-order accuracy. Furthermore, the fitted- and unfitted-mesh methods are shown to achieve similar accuracy and computational cost when their element sizes are matched.