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The propagation of elastic waves through heterogeneous, anisotropic media is considered. Appropriate ensemble averaging of the elastic wave equation leads to the Dyson equation which governs the mean response of the field. The Dyson equation is given here in terms of anisotropic elastic Green’s dyadics for the medium with and without heterogeneities. The solution of the Dyson equation for the mean response is given for heterogeneities that are weak. The formalism is further specified for the case of equiaxed cubic polycrystalline metals with a single aligned axis. The Green’s dyadics in this case are those for a transversely isotropic medium. Simple expressions for the attenuations of the shear horizontal, quasicompressional, and quasishear waves are given in terms of integrations on the unit circle. The derived expressions are limited to frequencies below the geometric optics limit, but give the attenuations in a direct manner. Comparisons with previous results are also discussed. It is anticipated that a similar approach is necessary for the study of wave propagation in complex anisotropic materials such as fiber-reinforced composites. In addition, the results are applicable to diffuse ultrasonic inspection of textured polycrystalline media.