First experience: Installed the app via web account a few hours ago. Came down to my Roku and signed in now - worked first time. I signed in via a mobile device (my iPhone) which worked first time.
The UI seems slightly slow, but I'm happy this even runs on my 2013 Roku 3. I'm watching BladeRunner 2049 on it now. Image quality and sound quality is GREAT. Well done for taking one step out of the walled garden, Apple!
Update: went to compare quality with plex and it hard rebooted my Roku when I quit the AppleTV app

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I had to reconfirm I was happy with data usage on re-start. It had remembered my sign-in and position in the movie though. This was quitting by pressing the back button repeatedly. Pressing home from a movie whilst playing worked.
* On my 1080p TV, Streaming is comparable to Plex streaming a HD file from a local server - as in great!
* Fast forward and rewind seem sluggish - for example, pressing play to stop rewind led to it rewinding a further minute before realising play had been pressed.
Still, for a first version, impressive! I'll use this, and it lets my Roku still stream
every service I use (Netflix, Amazon, NowTV, iPlayer, plex, and now iTunes)