Actually there is a simple 3D analog of the hexagonal grid, which occurs in crystal structures. Somewhat interestingly there are two variants, known in crystallography as Face-Centred Cubic and Hexagonal Close Packing.
http://en.m.wikipedia.org/wiki/Close-packing_of_equal_sphere...
In higher dimensions there are applications to broadband coding (you want the densest possible constellation).
Both types are important. The math is indeed more complicated for square meshes, but you can use the fact that the basis functions (is this the proper English translation of the German "Basisfunktionen"?) on square elements have more degrees of freedom to sometimes get better approximations.
Disclosure: I heard some advanced lectures about FEM intended for students who intended to write a diploma thesis about this topic.
In higher dimensions there are applications to broadband coding (you want the densest possible constellation).
http://mathworld.wolfram.com/HyperspherePacking.html