A variety of chemical and biological nonlinear excitable media, including heart tissue, exhibit vortices (spiral waves) that can anchor to nonexcitable obstacles. Such anchored vortices can be terminated by the application of high-frequency wave trains, as shown previously in isotropic excitable media. In this study, we examined the basic dependencies of the conduction velocities of planar waves and waves around curved obstacles as a function of anisotropy through numerical simulations of excitable media that mimic the fiber orientation in a real heart. We also investigated the unpinning of anchored spiral waves by high-frequency wave trains in an anisotropic excitable medium. Unlike the findings regarding the termination of spiral waves in isotropic excitable systems, we found a nonmonotonic relationship between the maximum unpinning period and the obstacle radius depending on the fiber orientation, where the formation of unwanted secondary pinned vortices or chaotic waves is seen over a wide range of parameters.