Grain growth microstructures are composed of polyhedral cells of varying sizes and shapes. These structures are simple in the sense that no more than three faces meet at an edge and no more than four edges meet at a vertex. Individual cells can thus be considered (with few degenerated exceptions) as simple, three-dimensional polyhedra. A question of recent interest involves determining the distribution of combinatorial types of such polyhedral cells. In this paper we introduce the terms \emph{fundamental} and \emph{vertex-truncated} types and demonstrate that most grains are of particular fundamental types, whereas the frequency of vertex-truncated types decreases exponentially with the number of truncations. This can be explained by the evolutionary process through which grain growth structures are formed, and in which energetically unfavorable surfaces are quickly eliminated. Furthermore, we observe that these grain types are `round' in a combinatorial sense: there are no `short' separating cycles that partition the polyhedra into two parts of similar sizes. A particular microstructure derived from the Poisson--Voronoi initial condition is identified as containing an unusually large proportion of round grains. This Round microstructure has an average of $14.036$ faces per grain, and is conjectured to be more resistant to topological change than the steady-state grain growth microstructure. ; Comment: 23 pages, 28 figures, 10 tables