I just recently updated computers as well and came across this same problem. It seems to be fixed for me now.
I first updated the final cut 7.0.3 with the pro applications update for 2010.
https://support.apple.com/kb/DL949?viewlocale=en_US&locale=en_US
I had a hard time updating this through mountain lion so I just downloaded it through the link rather than the app store.
Then I updated the ProApps QuickTime codecs 1.0 through the app store
Here is a link if that doesn’t work.
https://support.apple.com/kb/DL1396
Hopefully this works for you as well. I had completely spaced out on the updates and Lion or Moutain lion don’t seem to be the first to remind you about them.