The idea that periods should be subject to a "transcendental Galois theory" has been first advanced by Grothendieck, who sketched a beautiful (but extremely conjectural) relationship with his theory of motives and motivic Galois groups. The resulting Period conjecture is very closely related to the Kontsevich-Zagier period conjecture that you mention in your question. I could give more details, but instead I will recommend this short and beautiful [survey][1] together with this other [survey][2] for some recent developments.


  [1]: http://arxiv.org/abs/0805.2569
  [2]: http://user.math.uzh.ch/ayoub/PDF-Files/periods-GKZ.pdf