A new mechanism of superfast motion of resistive domains in anisotropic superconductors is suggested. For bicrystals or sandwiches of anisotropic superconducting materials, the superfast motion is shown to arise due to the additional Joule heating of the kink regions of a resistive domain by eddy currents associated with them. Conditions under which the resistive domain in an anisotropic superconductor moves with a velocity up to the Fermi velocity, , are discussed. As a result of the superfast motion, generation of electric oscillations of high amplitude with frequencies up to can take place.