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X-ray free-electron lasers based on superconducting accelerator technology deliver ultrashort photon pulses with unprecedented peak brilliance at high repetition rates. Continuous and noninvasive monitoring of the current profile of the electron bunches is essential for the operation and control of the accelerator. Longitudinal diagnostics based on coherent radiation have already shown their potential at various free-electron laser facilities, and the multi-GeV electron beams of x-ray free-electron lasers are powerful sources for the generation of broadband coherent diffraction radiation. We present noninvasive current profile measurements with a few femtoseconds resolution based on spectroscopy of coherent diffraction radiation in the frequency range 0.7–58 THz. The current profiles, reconstructed from the spectroscopic data with an advanced phase retrieval method, are compared with measurement results obtained with a transverse deflecting structure. For the first time, bunch-resolved current profiles have been recorded simultaneously to user operation at European XFEL for all bunches in the bunch train at MHz repetition rates.