We have used microwave dielectric relaxation spectroscopy to study the picosecond dynamics of five low-viscosity, highly conductive room temperature ionic liquids based on 1-alkyl-3-methylimidazolium cations paired with the bis((trifluoromethyl)sulfonyl)imide anion. Up to 20 GHz the dielectric response is bimodal. The longest relaxation component at the time scale of several 100 ps reveals strongly nonexponential dynamics and correlates with the viscosity in a manner consistent with hydrodynamic predictions for the diffusive reorientation of dipolar ions. Methyl substitution at the C2 position destroys this correlation. The time constants of the weak second process at the 20 ps time scale are practically the same for each salt. This intermediate process seems to correlate with similar modes in optical Kerr effect spectra, but its physical origin is unclear. The missing high-frequency portion of the spectra indicates relaxation beyond the upper cutoff frequency of 20 GHz, presumably due to subpicosecond translational and librational displacements of ions in the cage of their counterions. There is no evidence for orientational relaxation of long-lived ion pairs.