By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In $\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. Any other triangulation would give you a smaller minimum angle, so in this sense, triangulating $\mathbb{R}^2$ with equilateral triangles is the most regular way.

What's the best way to triangulate $\mathbb{R}^3$ so that the minimum solid angle of the tetrahedra is maximized?