We have studied the rotational and translational diffusion of the spin probe 4-hydroxy-2,2,6,6-tetramethyl-1-piperidinyloxy (TEMPOL) in five imidazolium-based room-temperature ionic liquids (RTILs) and glycerol by means of X-band electron paramagnetic resonance (EPR) spectroscopy. Rotational correlation times and rate constants of intermolecular spin exchange have been determined by analysis of the EPR line shape at various temperatures and spin probe concentrations. The model of isotropic rotational diffusion cannot account for all spectral features of TEMPOL in all RTILs. In highly viscous RTILs, the rotational mobility of TEMPOL differs for different molecular axes. The translational diffusion coefficients have been calculated from spin exchange rate constants. To this end, line shape contributions stemming from Heisenberg exchange and from the electron-electron dipolar interaction have been separated based on their distinct temperature dependences. While the Debye-Stokes-Einstein law is found to apply for the rotational correlation times in all solvents studied, the dependence of the translational diffusion coefficients on the Stokes parameter T/eta is nonlinear; i.e., deviations from the Stokes-Einstein law are observed. The effective activation energies of rotational diffusion are significantly larger than the corresponding values for translational motion. Effects of the identity of the RTIL cations and anions on the activation energies are discussed.