We describe and evaluate a numerical solution strategy for simulating surface acoustic waves (SAWs) through semiconductor devices with complex geometries. This multi-physics problem is of particular relevance to the design of SAW-based quantum electronic devices. The mathematical model consists of two coupled partial differential equations for the elastic wave propagation and the electric field, respectively, in anisotropic piezoelectric media. These equations are discretized by the finite element method in space and by a finite difference method in time. The latter method yields a convenient numerical decoupling of the governing equations. We describe how a computer implementation can utilize the decoupling and, via object-oriented programming techniques reuse independent codes for the Poisson equation and the linear time-dependent elasticity equation. First we apply the simulator to a simplified model problem for verifying the implementation, and thereafter we show that the methodology is capable of simulating a real-world case from nanotechnology, involving SAWs in a geometrically non-trivial device made of Gallium Arsenide.