The propagation of an acoustic wave generated by a δ-pulse unit point force in a three dimensional anisotropic medium is considered. A general expression for the retarded 3 × 3 elastodynamic displacement Green's function is derived for arbitrary anisotropic symmetry. The initial integration problem of four integrations is reduced to only one ϕ-integration problem combined with the problem of solving three algebraic equations. The evaluation of expressions of the Green's function for the behavior at infinity yields the radiation condition and describes the acoustic wave propagation and gives the possible distinguished propagation directions from the source point for outgoing acoustic waves (high symmetry directions of the crystal beside other distinguished directions). For high symmetry directions of the crystal this condition yields three outgoing singular acoustic waves, one with longitudinal and two with transverse polarization. For elastic isotropy of the medium we obtain the result known from literature [17] which was derived there by using other techniques. For hexagonal symmetry of the medium we derive closed-form expressions of the farfield waves for the high symmetry directions (i.e. for propagation directions parallel and in the plane perpendicular to the hexagonal c-axis).