Exact solutions are derived for the propagation of compressional elastic waves along an anisotropic circular cylinder having axial elastic symmetry (hexagonal elastic symmetry with the crystallographic axis coincident with the cylinder axis). Formal solutions are obtained for the displacement components, and a transcendental equation relating the phase velocity and the circumference‐to‐wavelength ratio is derived. The solutions are shown to reduce to the Pochhammer solution for the isotropic case. In the long wavelength extreme the velocity is dispersionless and is given in terms of the axial Young's modulus. In the short wave‐length limit the velocity again becomes constant, similar to the Rayleigh surface wave limit in the isotropic case.