We consider the frequency shifts of particles moving in metamaterials exhibiting the electromagnetically induced transparency (EIT) effect. The dramatic change of the material dispersion due to the EIT influences the conditions for signal propagation in the medium and has a significant impact on the Doppler effect, possibly leading to the optical control over this phenomenon. The dependence of the Doppler shift to the source frequency and velocity and radiation spectra on external parameters is examined. It was found that for source frequencies fitting a transparency window for a particular range of source velocities cutoffs appear, i.e., the radiation is not emitted. The analysis of cutoff dependence on the character of medium dispersion brings to light their physical interpretation. Our theoretical findings are proved analytically and confirmed by numerical simulations based on the finite-difference time-domain method.