The connectivity of a graph is an important measure of fault tolerance of the interconnection network represented by that graph. It is also closely related to the theory of network flow problems. In this paper, the authors determine the connectivity of the tensor product $G\times C$ of any connected graph $G$ and a cycle $C$. They obtain the exact value of the connectivity of $G\times C$ when $G$ is a bipartite graph or $C$ is an even cycle, and provide sharp bounds for the connectivity when $G$ is a non-bipartite graph and $C$ is an odd cycle. At the end of the paper, they show that the connectivity of the tensor product of $k$ odd cycles is $2k$.