We develop the complete theory for the collective plasmon modes of an interacting electron system in the presence of explicit mass (or velocity) anisotropy in the corresponding noninteracting situation, with the effective Fermi velocity being different along different axes. Such effective mass anisotropy is common in solid state materials (e.g., silicon or germanium), where the Fermi surface is often not spherical. We find that the plasmon dispersion itself develops significant anisotropy in such systems, and the commonly used isotropic approximation of using a density of states or optical effective mass does not work for the anisotropic system. We predict a qualitatively new phenomenon in anisotropic systems with no corresponding isotropic analog, where the plasmon mode along one direction decays into electron-hole pairs through Landau damping while the mode remains undamped and stable along a different direction.