A simple explanation I enjoy (not quite visual, but geometric in some sense) is the on the wiki using Fourier series, understanding it requires just a few steps:
1) You accept Parseval's identity: by changing between orthogonal basis, the energy (sum of absolute squared values) remains the same.
2) You find that the energy of the periodic function x mod pi is sum(1/n^2).
3) Evaluating the integral gives pi^2/6!
The square is neatly explained because it involves energies. I guess this is why the pattern works only for even powers.
The sum of the inverse fourth powers is pi^4/90. Sixth powers, pi^6/915. The pattern keeps going for even powers and not for odd.
I don't think there is a geometric proof. But if there were, it would be incredible.