Extensive deuterium NMR relaxation data are presented for two specifically deuterium labeled surfactants forming bicontinuous cubic phases with water. H-2 spin-lattice (R-1) and spin-spin (R-2) relaxation rates were measured over an extended frequency range from 2 to 60 MHz. The data are interpreted with an existing theoretical framework for spin relaxation in bicontinuous cubic phases, which takes its starting point in the description of bicontinuous phases using periodic minimal surfaces. We show that the theory succeeds in accounting for the data and that the defining parameters of the theory, correlation times and order parameters, are in agreement with related data in other surfactant phase situations. Specifically, we obtain the surfactant self-diffusion coefficient over the minimal surface in one unit cell and show that it is in agreement with the corresponding macroscopic NMR diffusion data. By measuring two additional NMR relaxation parameters for each carbon on the surfactant hydrocarbon tail, we demonstrate how order parameter and correlation time profiles can be obtained. Finally, we analyze published molecular dynamics trajectories for a bicontinuous cubic phase. The analysis provides further support for the theoretical framework used to interpret relaxation data.