A study is made of the dynamical behavior of an electron or exciton undergoing anisotropic hopping on a d-dimensional bond-disordered lattice. Starting with a master equation for the site probabilities, an exact equation of motion is obtained for the probability currents that flow along the bonds connecting nearest-neighbor sites. Unlike the original master equation, the equation of motion which couples the microscopic currents contains the randomly distributed hopping rates in a form which is strictly site diagonal. The simplification that results leads to a new and exact expansion for the diffusion tensor in powers of an appropriately defined single-bond t matrix. From the lowest term of this expansion, a frequency-dependent effective-medium theory for anisotropic solids is constructed. The theory is then used to study the vanishing transport anisotropy that occurs for an anisotropic random walk on an isotropically percolating lattice near the critical point
