We analyze controllability properties for a class of bilinear interconnected systems, consisting of networks of linear systems, where the coupling parameters act as control variables. We characterize the system Lie algebra of the resulting bilinear control systems. Necessary and sufficient conditions for accessibility are derived in terms of the underlying interconnection graph. Our results generalize earlier work by Brockett on controllability of bilinear output feedback systems, as well as recent work of Costello and Egerstedt on the control of information-exchange networks for distributed computing.