Unless you ask for the complex to have only one non-zero reduced homology group, you can realize any module you like using $H_1$ of a 2-complex, or $H_2$ of a 3-complex if you insist that the complex should be simply-connected. Given a $\mathbb{Z}G$-module $M$, take a presentation for $M$, i.e. an exact sequence $F_1\rightarrow F_0\rightarrow M\rightarrow 0$ in which each $F_i$ is a free $\mathbb{Z}G$-module. Now you can realize $F_0$ as the 2nd homology of a wedge of 2-spheres with $G$ acting freely except on the basepoint. You can attach a disjoint union of 3-balls permuted freely by $G$ (with $H_0$ isomorphic to $F_1$) in such a way that the cellular chain complex is just $F_1\rightarrow F_0\rightarrow 0\rightarrow \mathbb{Z}$, with the given map from degree 3 to degree 2. $H_3$ of this space is of course the kernel of the map $F_1\rightarrow F_0$, while $H_2$ is isomorphic to $M$.
With a wedge of 1-spheres and attached 2-cells everything works the same way (the Hurewicz theorem tells you that $\pi_1$ surjects onto $H_1$), except that the 2=dimensional complex will probably have fundamental group a lot bigger than the abelian group $M$.

This does not answer Greg's question, but it is related. You can realize any $\mathbb{Z}G$-module you that like as $H_1$ of a based 2-complex, or as $H_2$ of a 3-complex if you insist that the complex should be simply-connected. Furthermore, you can require $G$ to act freely on the complex except for fixing the base point.

Given a $\mathbb{Z}G$-module $M$, take a presentation for $M$, i.e. an exact sequence $F_1\rightarrow F_0\rightarrow M\rightarrow 0$ in which each $F_i$ is a free $\mathbb{Z}G$-module. Now you can realize $F_0$ as the 2nd homology of a wedge of 2-spheres with $G$ acting freely except on the basepoint.

You can attach a disjoint union of 3-balls permuted freely by $G$ (with $H_0$ isomorphic to $F_1$) in such a way that the cellular chain complex is just $F_1\rightarrow F_0\rightarrow 0\rightarrow \mathbb{Z}$, with the given map from degree 3 to degree 2. $H_3$ of this space is of course the kernel of the map $F_1\rightarrow F_0$, while $H_2$ is isomorphic to $M$.

With a wedge of 1-spheres and attached 2-cells everything works the same way (the Hurewicz theorem tells you that $\pi_1$ surjects onto $H_1$), except that the 2-dimensional complex will probably have fundamental group a lot bigger than the abelian group $M$.