A procedure to calculate the radiation spectrum emitted by an arbitrarily prepared Dirac wave packet is developed. It is based on the Dirac charge current and classical electrodynamic theory. Apart from giving absolute intensity values, it is exact in terms of relativistic retardation effects and angular dependence. We employ a laser driven free electron to demonstrate the advantages of our method as compared to traditional ones that merely rely on the Fourier transform of the dipole operator's expectation value. Classical reference calculations confirm the results obtained for the low-frequency part of the spectrum, especially in terms of the observed red-shifts, which clearly deviate from non-relativistic calculations. In the high-frequency part of the spectrum, we note appreciable deviations to the purely classical calculations which may be linked to quantum averaging effects.