We propose a novel and poweful methodology for three- dimensional (3D) grain growth modelling in both the anisotropic and the locally inhomogeneous cases, thus significantly generalizing previous related two-dimensional work. The fundamental modelling structures for the polycrystals and their behaviour are dynamically evolving Laguerre diagrams in the flat 3D torus. The weighted generating sites of such diagrams obey motion equations ensuing from system interface energy minimization with consequent evolution of the associated structures and resulting in elementary topological transformations thereof. We have implemented the models in state-of-the-art computer simulation codes and made large-scale runs assuming either constant (isotropic) or variable (anisotropic) interface specific energies. In both cases, the simulated evolution reproduced the main features of the normal grain growth process in polycrystalline materials, that is the grain growth power law, typical distributions of grain sizes and shapes, and the scaling behaviour in long-term regime. The simulated distribution of grain-to- grain misorientation and its evolution in a simple hypothetical case have also been obtained for the first time.