To understand the influence of organized structure of the ionic liquids on the rotational diffusion of a hydrogen bond donating ionic solute, reorientation times (tau(r)) of rhodamine 110 (R110) have been measured in 1-alkyl-3-methylimidazolium ([Rmim(+)]) based ionic liquids with anions tetrafluoroborate ([BF4-]) and hexafluorophosphate ([PF6-]). The viscosity (eta) was varied by changing the temperature (T) and also the alkyl chain length on the imidazolium cation (ethyl, butyl, hexyl, and octyl). It has been noticed that tau(r) versus eta/T plots contain two slopes corresponding to lower and higher values of eta/T for ionic liquids with [BF4-] as well as [PF6-] anions. For lower values of eta/T (<0.2 and <0.3 mPa s K-1, respectively, for [Rmim(+)] [BF4-] and [Rmim(+)][PF6-]), rotational diffusion of R110 follows Stokes-Einstein-Debye hydrodynamic theory with stick boundary condition due to specific interactions between the solute and the anions of the ionic liquids. In contrast, at higher eta/T, the rotational diffusion of the solute is faster than the stick predictions and this trend could not be explained by the quasihydrodynamic theories of Gierer-Wirtz and Dote-Kivelson-Schwartz as well. Diminishing hydrogen bonding interactions between the solute and the anions, which transpire as a consequence of the organized structure of the ionic liquids, are responsible for the observed behavior.