What are the semisimple $\mathbb{Z}$-modules?

If $M$ is a simple module over $\mathbb{Z}$ then it is generated by an element. In particular, $M$ is a cyclic group. The modules over $\mathbb{Z}$ are precisely abelian groups, and every non-cyclic abelian group has a non-trivial subgroup (which are therefore submodules). Thus, the semisimple modules are direct sum of prime cyclics.

A semisimple module is a direct sum of simple modules. So we just have to determine what are the simple modules. A module $M$ is simple if and only if it is of the form $\mathbb{Z}/I$, where $I$ is a maximal ideal. The maximal ideals of $\mathbb{Z}$ are of the form $p\mathbb{Z}$, where $p$ is a prime integer. So the general form of a semisimple module is $$ \bigoplus_{\text{$p$ prime}}(\mathbb{Z}/p\mathbb{Z})^{(\alpha_p)} $$ for any choice of cardinal numbers $\alpha_p$ (where $M^{(\alpha)}$ denotes a direct sum of $\alpha$ copies of $M$).