We explore the LQG control of a networked control system (NCS) in which a linear plant is controlled remotely over a network or other shared communication medium. The medium provides a limited number of simultaneous connections, so that only a subset of the plant's sensors and actuators may communicate with the controller at any one time, subject to known transmission delays. Instead of insisting on jointly optimal control and medium access policies, we reduce the infinity of possible access sequences down to those which preserve the stabilisability and detectability of the underlying plant, and are periodic. Our choice of communication and NCS model effect a kind of 'decoupling' of the LQG problem, in the sense that the medium access policy can be selected independently of the controller. This guarantees the existence of a stabilising LQG controller which is optimal for the communication policy of choice, and which is then combined with a delay compensator. We include simulations that illustrate our approach.