Elements are presented of the formulation and selected applications of the combined surface and volume integral-equation approach for finding pressure, displacement, and traction fields in composite objects consisting of piece-wise homogeneous regions characterized by different Lame's material parameters, as well as of highly inhomogeneous regions. The proposed elastodynamics surface/volume-based integral equations, in which the unknowns are volumetric pressure or displacement variables and surface displacement and traction fields on the interfaces, involve only weakly singular integral-equation kernels with at most 1/r singularity. The proposed method is found to provide well-convergent and accurate solutions for a large spectrum of problems, in particular those involving high-contrast material interfaces. The proposed method's accuracy and its capability (achieved through the (non-lossy FFT-based) Adaptive Integral Method matrix compression) of handling large-scale problems involving several millions of unknowns, is demonstrated through comparison with analytically constructed solution for a layered material sphere and in selected numerical simulations of propagation of acousto-elastic waves in human head and inner-ear region through air- and bone-conduction pathways. The proposed approach can be used in assessing effectiveness of noise-protection devices. This work is supported by a grant from the Air Force Office of Scientific Research.