An integral equation method is applied for the calculation of elastic wave fields in unbounded solids containing general anisotropic inclusions and voids. The domain of the integral equation involves the volume of the inclusions as well as the surface of the voids. In contrast to the conventional boundary integral equation method (BIEM), where the infinite medium Green's functions for both the matrix material and the inclusion material are needed, the present method does not require the latter. Since the elastodynamic Green's functions for anisotropic media are extremely difficult to calculate, the present method offers a definite advantage over methods based on boundary integral equation (BIE) alone. The newly developed mixed method takes full advantage of the volume integral equation method that is effective for problems with anisotropic inclusions and the BIEM that is effective for problems involving voids and cracks. In this paper, the mixed method is used to calculate the interaction of plane, time-harmonic elastic waves with an isotropic and an orthotropic cylindrical inclusion in absence or presence of a parallel cylindrical void in its vicinity, for waves incident normal to the cylinder axis. Numerical results are presented for the displacement and stress fields at the interfaces of the inclusions in a broad frequency range of practical interest. The new method is shown to be very accurate and efficient for solving this class of problems.