The temperature dependence of the molecular diffusion in monoolein/water systems is investigated at several levels of hydration. Using the proton/deuteron selectivity, field gradient NMR allows the simultaneous determination of the diffusion constants of both, lipid and water molecules in the various lamellar and non-lamellar phases. Due to the mesoscopic structure of the monoolein/water phases, the diffusion coefficients are interpreted as 'reduced' or 'effective' diffusion coefficients, and are related to the microscopic molecular displacements by a so-called 'obstruction factor'. Changes in the microscopic structure at the phase transition from the bicontinuous cubic phases to the inverse hexagonal phase are reflected in the obstruction factor of the monoolein diffusion coefficients. The reduction of the water diffusion coefficients is too high to be explained by an obstruction factor only, implying a mechanism of molecular motion, which strongly differs from that of bulk water. Experiments on samples prepared with isotopic labeled water (2H(2)O and H(2)(17)O) indicate a chemical exchange of protons between the water molecules and the lipid headgroups on a millisecond timescale.