Accurate simulation of seismic waves is of critical importance in a variety of geophysical applications. Based on recent works on staggered discontinuous Galerkin methods, we have developed a new method for the simulations of seismic waves, which has energy conservation and extremely low grid dispersion, so that it naturally provided accurate numerical simulations of wave propagation useful for geophysical applications and was a generalization of classical staggered-grid finite-difference methods. Moreover, it could handle with ease irregular surface topography and discontinuities in the subsurface models. Our new method discretized the velocity and the stress tensor on this staggered grid, with continuity imposed on different parts of the mesh. The symmetry of the stress tensor was enforced by the Lagrange multiplier technique. The resulting method was an explicit scheme, requiring the solutions of a block diagonal system and a local saddle point system in each time step, and it was, therefore, very efficient. To tailor our scheme to Rayleigh waves, we developed a mortar formulation of our method. Specifically, a fine mesh was used near the free surface and a coarse mesh was used in the rest of the domain. The two meshes were in general not matching, and the continuity of the velocity at the interface was enforced by a Lagrange multiplier. The resulting method was also efficient in time marching. We also developed a stability analysis of the scheme and an explicit bound for the time step size. In addition, we evaluated some numerical results and found that our method was able to preserve the wave energy and accurately computed the Rayleigh waves. Moreover, the mortar formulation gave a significant speed up compared with the use of a uniform fine mesh, and provided an efficient tool for the simulation of Rayleigh waves.