It is definitely not known if $\zeta(3)/\pi^3$ is rational or not. By the way, there is [a paper of Felder and Willwacher][1] where they prove that the weight of a certain graph appearing in [Kontsevich's formality quasi-isomorphism][2] is, up to a rational, $\zeta(3)/\pi^3$. The question whether Kontsevich's quasi-isomorphism is defined over $\mathbb{Q}$ or not, is still open. If the answer to this question would be "yes", then the [associator defined by Alekseev and Torossian][3] would have rational coefficients... and that would definitely be a great result!

Among the main recent advances concerning rationality of zeta values, there are the works of [Tanguy Rivoal][4] and [Wadim Zudilin][5]. One of the most advanced results is that there is at least one irrational in $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$. 


  [1]: http://arxiv.org/abs/0808.2762
  [2]: http://arxiv.org/abs/q-alg/9709040
  [3]: http://arxiv.org/abs/0906.0187
  [4]: http://math.univ-lyon1.fr/homes-www/rivoal/articles.html
  [5]: http://wain.mi.ras.ru/publications.html