We derive exact expressions for effective elastodynamic properties of two-phase composites in the long-wavelength (quasistatic) regime via homogenized constitutive relations that are local in space. This is accomplished by extending the ‘strong-contrast’ expansion formalism that was previously applied to the static problem. These strong-contrast expansions explicitly incorporate complete microstructural information of the composite via an infinite set of \(n\)-point correlation functions. Utilizing the rapid-convergence properties of these series expansions (even for extreme contrast ratios), we extract accurate approximations that depend on the microstructure via the spectral density, which is easy to compute or measure for any composite. We also investigate the predictive power of modifications of such approximation formulas postulated elsewhere (Kim and Torquato 2020 Proc. Natl Acad. Sci. 117 8764) to extend their applicability beyond the quasistatic regime. The accuracy of these nonlocal microstructure-dependent approximations is validated by comparison to full-waveform simulation results for certain models of dispersions. We apply our formulas to a variety of models of nonhyperuniform and hyperuniform disordered composites. We demonstrate that hyperuniform systems are less lossy than their nonhyperuniform counterparts in the quasistatic regime, and stealthy hyperuniform media can be perfectly transparent for a wide range of wavenumbers. Finally, we discuss how to utilize our approximations for engineering composites with prescribed elastic wave characteristics.