I have exactly the same problem. It is a brand new airport express of the last generation. The device is used for audio playback and it is wired, wifi is switched off. As long as I'm streaming music everything is ok, it keeps working forever. When the device is not used for a while it loses connection. The light is still green and flashes so now and then but neither iTunes nor the airport utility can find the device on the network anymore. When I disconnect the network cable of the airport for one second and then connect it again the device is visible again. I use 2 other airports of older generations as well (not for streaming but for wifi extension) they don't have that problem.