The purpose of this work is to draw attention to several differences between wave propagation in dissipative anisotropic media and purely elastic anisotropic media. In an elastic medium, the wavefront is defined as the envelope of the family of planes that makes the phase of the plane waves zero. It turns out that this definition coincides with the wavefronts obtained from the group and energy velocities, i.e., the three concepts are equivalent. However, for plane waves traveling in dissipative anisotropic media these concepts are different. Despite these differences, the velocity of the envelope of plane waves closely approximates the energy velocity, and therefore can represent the wavefront from a practical point of view. On the other hand, the group velocity describes the wavefront only when the attenuation is relatively low, i.e., for Q values higher than 100. The values of the different velocities and the shape of the wavefront are considerably influenced by the relative values of the attenuation along the principal axes of the anisotropic medium. This means that the anisotropic coefficients in attenuating anisotropic media may differ substantially from the corresponding elastic coefficients. Moreover, it is shown that the usual orthogonality properties between the slowness surface and energy velocity vector and the wavefront and wavenumber vector does not hold for dissipative anisotropic media.