A three-dimensional (3-D) generic hybrid model is developed for the simulation of elastic waves in applications in nondestructive evaluation (NDE) that efficiently links different solution strategies but, crucially, is independent of the particular schemes employed. This is an important step forward in facilitating rapid and accurate large-scale simulations, and this advances the two-dimensional (2-D) generic hybrid methodology recently developed by the authors. The hybrid model provides an efficient and effective tool for creating highly accurate simulations that model the wave propagation and scattering, enabling the interpretation of inspection data; the new methodology is verified against other numerical simulations. Furthermore, its deployment to simulate wave reflection from side-drilled holes (SDHs), comparing the results with experimental measurements, provides a realistic demonstration as well as further validation.