Yes, it sounds like you are rendering to a file format or codec that the new machine doesn't have installed.
What is the format and codec settings you use for your AE exports? What OS do you use? What hardware?
Format is QuickTime - channels RGB & Alpha.
The old one was in avi format, but I couldn't find a codec that could do that in avi (must admit I was playing a bit with options, not knowing what to do) - so don't know what codec it was before.
OS is windows 7 - PC Core I7 16 GB Ram
Have you installed QuickTime? If not, install QuickTime 7.
You didn't mention which Quicktime codec you've tried.
Like Szalam says, ensure you have Quicktime 7 installed.
You can render Quicktime codecs like Animation, PNG and Uncompressed with an alpha channel.
You can also render an Uncompressed AVI with an alpha channel.