We present an efficient technique to simulate seismic wave propagation in structures including basins using localized discontinuous grids in combination with locally variable timesteps. The technique uses a 3D fourth-order, staggered-grid, finite-difference method based on the velocity-stress formulations of the elastodynamic equations. Discontinuous grids have 3D blocky form; thus, the boundary between the two adjacent regions of different grid spacings could be extended to the position of the Earth's free surface. As an example of application, ground motions in a small-scale basin in the Hongseong area, Korea, is simulated considering a hypothetical source of the 7 October 1978 Hongseong, ML 5, earthquake. The results show that the method can achieve distinguished reductions of computational memory and thus CPU time for models of the localized 3D structures compared with methods on grids with constant spatial and temporal steps.