Temperature-dependent fluorescence anisotropies of a nonpolar solute 9-phenylanthracene (9-PA) have been measured in 1-alkyl-3-methylimidazolium-based ionic liquids with anions such as bis(trifluoromethylsulfonyl)imide ([Tf2N-]), tris(pentafluoroethyl)trifluorophosphate ([FAP(-)]), tetrafluoroborate ([BF4-]), and hexafluorophosphate ([PF6-]) to find out if the organized structure of the ionic liquid has a bearing on solute rotation. Analysis of the experimental: data using the Stokes-Einstein-Debye hydrodynamic theory indicates that there is no significant variation in the solute-solvent coupling constants (C-obs) with an increase in the length of the alkyl chain on the imidazolium cation for the ionic liquids with [Tf2N-] and [FAP(-)] anions. However, in the case of ionic liquids with [BF4-] and [PF6-] anions, the rotation of 9-PA for a given viscosity at constant temperature becomes progressively faster and C-obs decreases by a factor of 2.4 from ethyl to octyl derivatives. Quasihydrodynamic theories of Gierer-Wirtz, and Dote-Kivelson-Schwartz could not account for the significant decrease in the C-obs, values. The observed behavior has been rationalized in terms of the organized structure of the ionic liquids having [BF4-] and [PF6-] anions, which results as a consequence, of the high charge-to-size ratio of these anions compared to [Tf2N-] and [FAP(-)].