It's almost counter-intuitive the way this works. You have to actually "unpair" the remote that it is paired to - then once no specific Apple TV remote is paired, all remotes (any remote, including standard TV remotes) will work!
The step-by-step would be:
Using a remote that is currently paired navigate to General under settings.
Go to Remotes.
Select "Unpair Apple Remote"
Once unpaired, simply back out of all the menus using the menu button to see that it does work. Then try on your other remote(s). It/they will now also work.
You can even use the Learn Remote feature to use a standard TV remote. I use my Directv remote and it works great.
The ONLY thing you need pairing for is if you want only one remote to be capable of controlling the ATV. So therefore unpair any remote that's paired, and you'll be golden.