We develop new low-dissipation central-upwind (LDCU) schemes for nonlinear elasticity equations in heterogeneous media. In general, central-upwind schemes belong to the class of finite-volume Godunov-type schemes, which consist of three steps: reconstruction, evolution, and projection onto the original grid. In our new method, the evolution is performed in the standard way by integrating the system over the space-time control volumes. However, the reconstruction and projection are performed in a special manner. First, we take into account the fact that the conservative variables (strain and momentum) are discontinuous across the material interfaces, while the flux variables (velocity and strain) are continuous there: we therefore reconstruct the flux variables. Second, we use a special projection recently introduced in [A. Kurganov and R. Xin, J. Sci. Comput., 96, 2023] to complete the derivation of the LDCU scheme. Our numerical experiments demonstrate that the developed schemes are capable of accurately resolving waves with dispersive behavior that over a long period of time evolve into solitary waves.