You really do not want to migrate everything as-is, especially for such a volume. From experience, I'm willing to bet that a good chunk of that 3TB is accessed only rarely, if at all, but it still has an impact on site browsing performance. Identify your larger sites / lists as prime candidates for cleanup. What type of documents do you have? What's taking the most space?
- Clean up your existing content (delete, re-structure, archive)
- Design your new structure
- Move the content
I'm going to post a link from a colleague related to this topic: Migration from SharePoint 2007 to SharePoint 2010 – A real life scenario - Part 3
Self-promoting link : Check out Sharegate to migrate your content. It has no volume limit on content migration (I work for the makers of Sharegate).
If you want to migrate whole sites / structures / farms, there are plenty of tools that will do that too from AvePoint, Axeler, Idera, Metavis, Tzunami.
Also semi-self promoting (or rather promoting my field of work :D) : If you do not have strong expertise in house with SP2010 and SharePoint in general, hire help. Someone to help you with the move and with the differences between 2007 and 2010 will be a good investment. I have seen many customers call us after the move, helpless with an expensive farm used at 10% of it's features.
Trust me, migration is really one of those cases where preparation is half the battle. SharePoint farms are expensive to keep up. SharePoint SQL storage (particularly high-availability storage) is expensive. This is a good opportunity to do this right and maximise your ROI.