The eikonal method is extended to waves with several components propagating in inhomogeneous anisotropic media. Formulas are derived for the motion of wave packets, the change in amplitude along a ray path, and the corrections due to diffraction. The method is applied to pure magnetohydrodynamic disturbances, and the problem of computing ray paths in the ionosphere is discussed in some detail. One result is that the fraction of the energy of an isotropic disturbance that eventually gets as low as 200 km drops from 0.99 at 200 km to 0.2 at 450 km, rises again to 0.99 at 700 km, drops to 0.5 at 3000 km, and continues dropping at greater altitudes. Energy trapped between 200 km and 700 km oscillates around 450 km with period 8.5 sec (at the geomagnetic equator). The eikonal method is also applied to a very general problem involving coupled magnetohydrodynamic, electrodynamic, and acoustic modes.