The effects of anisotropy upon elastic wave propagation along a fluid-filled cylindrical borehole are determined. The wave equation is solved in the frequency-wavenumber domain with a variational method, and the solution yields the phase velocities, group velocities, pressures, and displacements for the normal modes. These properties are studied for two cases: a transversely isotropic model for which the borehole has several different orientations with respect to the symmetry axis and an orthorhombic model for which the borehole is parallel to the intersection of two symmetry planes. The normal modes for these two cases show several significant effects which do not exist when the solid is isotropic or transversely isotropic with its symmetry axis parallel to the borehole: 1. The phase velocities for the quasi-pseudo-Rayleigh, both quasi-flexural, and both quasi-screw waves do not exceed the phase velocity of the slowest qS-wave. (The phase velocities of the leaky modes, which were not investigated, will exceed this threshold.