Let $R=\mathbb{Q}[x,y,z]$, then every simple $R$-module $M$ is finite dimensional over $\mathbb{Q}$.
Had this been over $\mathbb{C}$ (complex field), it would have been rather easy. I have tried to use a theorem which says simple modules over $R$ is isomorphic to $R/I$, where I is a maximal regular ideal. But I don't understand regular ideals all that well. (For example, $Q[x,y]/(xy-1) \simeq Q(y)$, but $Q(y)$ is not finite dimensional over $Q$.)