OK, I've finished checking this now.
As expected, changing the HDMI lead didn't solve the problem. I have now swapped the two Apple TV's over and the result is as I imagined it would be.
When either Apple TV is plugged into by Sony Bravia 22EX302 it suffers from this problem. When either is plugged into the 40EX403 they are OK (one has been OK for days and the other for about 2 hours now).
From this test and what I have read on various forums, I think that we can be fairly sure that the problem is some sort of handshaking issue between the new Apple TV and some TV's. My guess is that if you suffer from this problem with a particular TV then changing cables or even returning the Apple TV for a replacement will have no effect because it is a software issue.
This presents those of us with the issue with real problems. We are reliant on an update coming along to fix the issue, but Apple could probably legitimately point to the fact that it works on the vast majority of TVs to say that it must be an issue for the TV manufacturer. The TV manufacturer can say that as most other HDMI products work it must be down to Apple.
To try to get some movement here, and to help Apple to identify the specific problem, I think it may be worth people noting down the exact TV they are using and whether they are suffering from the problem. The easier we make it for Apple to identify the issue the more chance there is that they can come up with a patch.
I'll start off:
Sony Bravia 40EX403U - Works fine
Sony Bravia 22EX302U - Suffers from this problem.
By the way, I do probably have the opportunity to return the TV at the moment and get a different model. However, I like the TV and it's good that both my sets have similar menus/controls etc. From peoples experience, and if anyone has knowledge of these sort of connections, what are the chances realistically of it getting sorted out? If a solution doesn't come along then I really need to try to return the TV.