Our anisotropic rate equation model outlines the relationship between the relaxation dynamics in a four-level solid-state laser and its anisotropic gain properties. Anisotropic pump rates and stimulated emission cross-sections were included to account for specific atom orientations in the gain material. The model is compared with experimental measurements of two relaxation oscillation frequencies which are related to the anisotropic atom-laser interaction in orthogonally polarized dual-mode lasers. The model predicts that crystal orientation and pump polarization affect the laser operation characteristics, as found experimentally. The gain anisotropy influences the fast laser dynamics, as in single-mode relaxation oscillations.