The detailed investigation of the local mobility in a set of dried imidazolium-based ionic liquids (1-butyl-3-methylimidazolium) in a wide temperature range and varying anions (BF4-, I-, Cl-, Br-, NO3-, TfO-) is presented. The measurements of temperature dependencies of the spin-lattice relaxation times of H-1 and C-13 nuclei are motivated by the need to obtain a fundamental characterization of molecular mobility of the substances under study, namely, to estimate the correlation times, tau(c), for the motion of individual molecular groups. In particular, it follows from obtained results that the mobility of the hydrocarbon "tail" is higher (smaller tau(c)) than that of the imidazole ring, and this expected tendency is quantified. The effect of the influence of an anion type on the cation mobility is also analyzed.