We model the grain boundary tracer diffusion problem by constructing a 3D structure consisting of cubic grains each of equal volume. We build the structure in such a way that no four cubes have a common edge. It is shown that the transition point between Harrison Type-A and Type-B kinetics regimes occurs at a diffusant diffusion length roughly an order of magnitude smaller than for the extensively studied case of parallel grain boundary slabs. For two dimensional squares the transition point occurs at a diffusion length roughly a factor of five smaller than for parallel grain boundary slabs. Thus we can draw the conclusion that dimensionality and geometric shape are both important factors in the parametric analysis of the grain boundary diffusion problem.