In the present work we performed low-frequency mechanical spectroscopy experiments to measure the mechanical modulus of two ionic liquids and its variation during the main phase transitions occurring by varying the temperature, in the both liquid and the solid states. The liquids share the same anion, the bis(trifluoromethanesulfonyl)imide, and present different cations, 1-butyl-1-methylpyrrolidinium and 1-allyl-3-H-imidazolium. A thermally activated relaxation process is found in the liquid phase and is analyzed in terms of a modified Debye model. The obtained parameters provide indications about the nature and the mechanism giving rise to the peak, which is attributed to the ions motion by means of hopping processes. Moreover, density functional calculations were performed, and the comparison with the analysis of the experimental data suggests that the anion conformers are likely to be involved in the different configurations among which the ions can rearrange.