The Chinese Remainder theorem is usually thought of as an isomorphism of rings, not just of cyclic groups. In this regard it has a vast generalization:
Theorem (Ideal-theoretic CRT): Let R be a commutative ring, and let $I_1,\ldots,I_n$ be a finite set of ideals in $R$ which are pairwise comaximal: for all $i \neq j$, $I_i + I_j = n$R$. Then $I_1 \cap \ldots \cap I_n = I_1 \cdots I_n$ and the natural homomorphism
$R/I_1 \cdots I_n = R/I_1 \cap \ldots \cap I_n \rightarrow \bigoplus_{i=1}^n R/I_i$
is an isomorphism. (See e.g. Theorem 41 on p.31 of http://math.uga.edu/~pete/integral.pdf.)
One could also think of $\mathbb{Z}/n\mathbb{Z}$ as a $\mathbb{Z}$-module, and then the CRT decomposition is a special case of primary decomposition for $R$-modules. In general rings, primary decomposition is somewhat complicated (e.g. it need not be unique), but for finitely generated torsion modules over a PID there is a straightforward analogue.
Finally, thinking about it in terms of groups, CRT has the following generalization: a finite group is nilpotent iff each Sylow $p$-subgroup is normal and $G$ is the direct product of its Sylow $p$-subgroups. There are Sylow decompositions in certain other group-theoretic contexts as well, e.g. nilpotent profinite groups.
