We consider the radiation from oscillating electric and magnetic dipoles moving with constant velocity directed parallel or antiparallel to the velocity of the surrounding medium. It is assumed that the medium in its rest frame is isotropic and has no spatial dispersion. We obtain expressions for the spectral density of the radiated power. In the case of a nondispersive medium, algebraic expressions for the total radiated power in the regime of "subluminal relative motion" are also obtained. In particular, it is shown that the energy loss of a source is negative if it moves in the direction of the superluminal motion of the medium and the source velocity is somewhat smaller than the medium velocity. It is noted that this phenomenon takes place for a smaller difference between the velocities of the source and the medium compared with a similar phenomenon for nonoscillating sources.