In this paper we study the antiplane problem of concentrated point force moving-With constant velocity and oscillating with constant frequency in unbounded homogeneous anisotropic elastic medium. The explicit representation of the elastodynamic Green's function is obtained by using Fourier integral transform techniques for all rates of source motion as a sum of the integrals over the finite interval. The dynamic and quasistatic components of the Green's function are extracted. The stationary phase method is applied to derive an asymptotic approximation at the far wave field. The simple formulae for Poynting energy flux vectors for moving and fixed observers are presented too. It is shown that the motion brings some differences in the far field properties, such as, for example, fast and slow waves appearance under superseismic motion and modification of the wave propagation zones and their numbers. The case of isotropic medium is considered separately. For isotropic material all main formulae are obtained in explicit forms.