The results of computation of main characteristics (slowness surface, ray surface, phase velocity and polarization vector) of three body seismic waves propagating in a homogeneous anisotropic elastic medium of arbitrary symmetry are presented. The study was directed to the correct solution of eigenvalues-eigenvectors problem and shear wave recognition in numerical algorithms for arbitrary direction of propagation in anisotropic media. The effects of wave propagation and their connection with the symmetry elements were investigated in anisotropic media for all basic types of symmetry.