We have measured the mean square displacement of a particle (MSD) and mean product of displacement of a particle and that of another particle (x-MSD) in two-dimensional (2d) and three-dimensional (3d) colloidal crystals for the first time using digital video microscopy. These (x-)MSDs have been compared to an overdamped bead-spring lattice model with effective viscous drag gamma (eff) (the OBS-gamma (eff) model). The observed 3d system contained ordered structures created by sedimentation equilibrium adjacent to the internal cell wall and was regarded as the (1 1 1) surface of a face-centered cubic lattice. The observed 2d system was a single layer of hexagonal-like ordered structures which were generated in a certain region of the space between the flat surface of the internal cell wall and a convex lens. In the observed time region, the MSDs for 2d system and 3d system were in good agreement with the theoretical MSDs for 2d system and 3d system, which were predicted to logarithmically diverge and to converge, respectively. The observed x-MSDs for the 2d system were in disagreement with theoretical ones in short time behavior. For the 3d system, the disagreement between the theory and the experimental results were found in the amplitudes as well as in the short time behavior. It was assumed that these differences were caused by the hydrodynamic coupling which was not fully incorporated in the OBS-gamma (eff), and an improved treatment, the OBS-cutoff model was introduced. The OBS-gamma (eff) elongates the relaxation time of each mode with the constant ratio of gamma (eff)/gamma, while the OBS-cutoff assumes the relaxation time of the modes to be infinity if the wave number of a mode is larger than a certain cutoff wave number. For the 3d system, the MSD and x-MSD for nearest neighbors obtained from the OBS-cutoff were in excellent agreement with the observed ones. In addition, it was found that the hydrodynamic interaction was the dominant factor on the short time behavior of (x-)MSD.