Let $D$ be a Dedekind domain and $G$ be a periodic Abelian-by-finite group. In this paper we study $DG$-modules in which every factor-module, apart from the trivial one, is $DG$-Artinian. In particular we prove that such modules cannot be $D$-periodic and that $G$ must be subject to some restrictions. Finally, we give a detailed description of such modules when $G$ is periodic Abelian and the spectrum of $D$ is infinite.