We report a high energy-resolution neutron backscattering study to investigate slow motions on nanosecond time scales in highly oriented solid-supported phospholipid bilayers of the model system deuterated 1,2-dimyristoyl-sn-glycero-3-phosphatidylcholine, hydrated with heavy water. Wave-vector-resolved quasielastic neutron scattering is used to determine relaxation times tau , which can be associated with different molecular components, i.e., the lipid acyl chains and the interstitial water molecules in the different phases of the model membrane system. The inelastic data are complemented by both energy-resolved and energy-integrated in situ diffraction. From a combined analysis of the inelastic data in the energy and time domains, the corresponding character of the relaxation, i.e., the exponent of the exponential decay, is also determined. From this analysis we quantify two relaxation processes. We associate the fast relaxation with translational diffusion of lipid and water molecules while the slow process likely stems from collective dynamics.