The pulsed field gradient NMR method has been used to determine self-diffusion coefficients of surfactant molecules, solubilized hydrocarbons, and D2O solvent in ternary N-alkyl-N,N-dimethylamine oxide/alkane/D2O surfactant systems. In the micellar phase, diffusion is governed by hydrodynamic transport of the micelles, supplemented by an exchange of solubilized hydrocarbon upon micellar collisions. This model is tested by variations in both the surfactant chain length and in the size of the hydrocarbon molecules. In the cubic ("ringing gel") phase, the solvent still exhibits values of the diffusion coefficients that are typical for motion in a continuous water phase, with the microemulsion droplets acting as obstacles. Mobilities of the surfactant in the gel state are low and have been determined only for the surfactant with the shortest chain length (C-12-DMAO). Exchange of hydrocarbon between micellar entities in the gel is found to occur by a hopping process; the associated rate decreases with the surfactant chain length. The latter process is restricted to domains with sizes on the order of a few microns.