Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse time migration and full-waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we have developed a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid (FSG) with a mimetic FD (MFD) (FSG + MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation in which we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG + MFD numerical method has a computational complexity comparable with Cartesian scenarios using a similar FSG + MFD numerical approach. Numerical examples demonstrate that our solution can simulate anisotropic elastodynamic field solutions on irregular geometries; thus, it is a reliable tool for anisotropic elastic modeling, imaging, and inversion applications in generalized computational domains including handling free-surface topography.