A novel algebraic method for the simultaneous estimation of MIMO channel parameters from channel sounder measurements is developed. We consider a parametric multipath propagation model with discrete paths where each path is characterized by its complex path gain, its directions of arrival and departure, time delay, and Doppler shift. This problem is treated as a special case of the multidimensional harmonic retrieval problem. While the well-known ESPRIT-type algorithms exploit shift-invariance between specific partitions of the signal matrix, the rank reduction estimator (RARE) algorithm exploits their internal Vandermonde structure. A multidimensional extension of the RARE algorithm is developed, analyzed, and applied to measurement data recorded with the RUSK vector channel sounder in the 2 GHz band.