Molecular dynamics (MD) simulations have been used to study the translational and rotational relaxation of model spherical nanocolloidal particles in solution at infinite dilution. The solvent was modelled at the molecular level. Simulations were carried out with two types of model nanocolloidal particle, one that was smooth (`structureless') and the other built from a cluster of atoms (`rough'). Both types had variable diameter, , compared to that of the solvent molecule, . The Weeks-Chandler-Andersen (WCA) interaction between the colloid and the WCA solvent molecules was used. Nanocolloidal particles that were up to an order of magnitude larger than those of the solvent molecules were simulated. The effects of the relative solvent and colloidal particle mass density, and colloid size on the translational and rotational self-diffusion coefficients were investigated. At liquid-like number densities the translational, D, and rotational, , self-diffusion coefficients for the nanocolloids of all sizes were statistically independent of the ratio of colloidal to solvent particle mass density for the the values, up to , explored. As solvent number density decreased, the translational self-diffusion coefficients of the colloidal particles showed more evidence than the rotational self-diffusion coefficients of a decrease with increasing colloid particle density. Both D and decreased with increasing size of the colloidal particle in close agreement with the classical solutions, the Stokes-Einstein and Stokes-Einstein-Debye relationships respectively. Differences in the translational diffusion coefficients of smooth and rough colloidal particles were not statistically significant at , but at the D were lower for the rough particles. Reorientational motion occurred by small-step diffusion.