Although this question already has some great answers, I thought it might be nice to note that if we drop the commutativity condition then in the case of integral group rings this question has an answer of non-abelian torsion-free polycyclic groups.

In particular, in the paper of Stafford http://www.numdam.org/article/CM_1985__54_1_63_0.pdf, Theorem 2.12 constructs a (right) stably-free non-free module over the integral group ring of any non-abelian torsion-free polycyclic group. 
We also have that projective modules are stably-free as polycyclic groups are solvable, and hence satisfy the Farrel-Jones conjecture.

Finally, virtually polycyclic groups are the only known examples of Noetherian integral group rings (or group rings in general), so fit this condition quite nicely.