This work presents a new Brownian dynamics simulation method of translational diffusion on curved surfaces. This new method introduce any implicit defined surface into the stochastic differential equation describing Brownian motion on that surface. The surface curvature will thus enter the force term (A) in the stochastic differential equation dX(t)=A(X-t)dt+B(X-t)dW(t) describing an Ito process. We apply the method calculating time correlation functions relevant in nuclear magnetic resonance (NMR) relaxation and translational diffusion studies of cubic phases of lyotropic systems. In particularly we study some bicontinuous cubic liquid crystalline phases which can be described as triply periodic minimal surfaces. The curvature dependent spin relaxation of the Schwarz-P minimal surface is calculated. A comparison of relaxation is made with the more complex topology of the Neovius surface which is another minimal surface in the same space group, and with parallel displacement of the minimal surface which thus results in a nonminimal surface. The curvature dependent relaxation effects are determined by calculating the translational diffusion modulated time-correlation function which determine the relaxation rates of a quadrupole nuclei residing in the water-lipid interface. The results demonstrates that spin relaxation data can provide quantitative information about micro-structure of biocontinuous cubic phases and that it is sensitive to the topology of the surface and to parallel displacement of the model surface. Consequently, spin relaxation may be used as a complement to x-ray diffraction in order to discriminate between different microstructures. It is concluded that fast and accurate computer simulations experiments is needed to be able to interpret NMR relaxation experiments on curved surfaces.