Additional for my answer above:
Thanks Glycerine. Do you think there would be any performance improvements if it was compressed into an older format? Or even just a SWF? There is no audio, so it's just an animated background really. – - Matt Brailsford
I think a newer format would be better - if you can do FP10, then again, you'll be able to utilise the user GPU, if your working in CS3, best to go for FP9.5.
Ensure your stage objects are cached for bitmap if your using large vectors
http://www.adobe.com/devnet/flash/articles/bitmap_caching_print.html
This ensures any heavy animation (even animation we regard as light) will run smoother because there turned into pixel data as opposed to complicated vector information. Its a small fix but it may work.
Try and target the AS3 engine as well. Even if your not using code. I keep saying it runs better than the as2, as1 engine with arguments from people but I'm sure you'll find your favourite camp.
If you have very large images scaled down, use a smaller form factor by photoshoping then to a smaller size. This will not only improve rendering speeds, but also swf file size.
Try those.