We consider the calculation of electromagnetic fields generated by a point charge passing through an anisotropic vacuum chamber of round or rectangular cross section with translational symmetry in the beam direction. The anisotropy causes a frequency shift and can have a significant effect on modal coupling. It must be accounted for in the design and analysis of accelerating structures. In this paper, field matching and finite-difference techniques for the anisotropic waveguides are developed. Additionally a method that mixes the field matching and the finite-differences is described. The numerical algorithms are implemented in a computer code and cross-checked on several examples. The impact of the anisotropy along different coordinate axes on the resonance frequency shifts is computed and analyzed.