Temperature-dependent fluorescence anisotropies of two organic solutes, 2,5-dimethyl-1,4-dioxo-3,6-diphenylpyrrolo[3,4-c]pyrrole (DMDPP) and rhodamine 110 (R110), have been measured in 1-butyl-3-methylimidazolium ([bmim(+)])-based ionic liquids containing the anions hexafluorophosphate ([PF6-]), bis (trifluoromethylsulfonyl)imide ([Tf2N-]), tetrafluoroborate ([BF4-]), trifluoromethanesulfonate ([TfO-]), and nitrate ([NO3-]). This data has been used in conjunction with the recently published results (Dutt, G. B. J. Phys. Chem. B 2010, 114, 8971) for the same solutes in [bmim(+)] tris(pentafluoroethyl)trifluorophosphate ([FAP(-)]) to understand the influence of various anions on solute rotation. The boundary condition parameter C-obs, which has been obtained from the analysis of the data using Stokes-Einstein-Debye hydrodynamic theory, for the neutral solute DMDPP is more or less the same in all the ionic liquids. Moreover, C-obs values are close to the predictions of slip boundary condition, which indicates that solvent viscosity alone governs the rotation of DMDPP. In contrast, for R110, which experiences specific interactions with the anions of the ionic liquids, the C-obs values are close to stick hydrodynamics. It has also been noticed that the C-obs values vary with the nature of the anion and this variation correlates with the hydrogen bond basicities of the anions of the ionic liquids.