Happy update!
After repeatedly trying to get TG4 to work properly on my Kindle Fire, I figured the browser had to be to blame. If you have a Kindle, you know it comes with the Silk browser, and Amazon really tries to make it difficult for you to get a different one (I couldn't even download Chrome from Chrome's own website!).
Turns out there is a workaround, and this site tells you how to do it:
http://www.thinkcomputers.org/how-to-in ... e-fire-hd/I'm now happily watching TG4 in full-screen on my Kindle Fire (which has a beautiful picture, by the way!)!
Redwolf