This talk deals with Cox rings of Calabi-Yau varieties X which are general anticanonical hypersurfaces in smooth toric Fano varieties Z. We present two complementary results, formulated in terms of primitive pairs of the anticanonical polytope of Z. The first gives combinatorial conditions ensuring that X is a Mori dream space and provides an explicit presentation of its Cox ring. The second shows that certain relations among primitive pairs force Bir(X) to be infinite, hence X is not a Mori dream space. As an application, we show that for Calabi-Yau hypersurfaces in dimensions two and three, either the Cox ring is finitely generated or the birational automorphism group is infinite. In the K3 case, where the Mori dream classification was already known via lattice theory, our approach gives a combinatorial interpretation together with explicit Cox ring presentations in the Mori dream cases. This is joint work with Antonio Laface and Luca Ugaglia.