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 almost exactly fits the 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.

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.