From the Organizer menu use:
File >> Get Photos & Videos >> From Files & Folders
Navigate to My Pictures (or your root folder if different)
Make sure you check the box to include all sub-folders
Then click Open or Get Media
Wait for thumbnails to be generated and accept (select all) Keyword or metadata suggestions.
... should I be looking at a different system altogether?
If your entire organizational goals are to continue to use your "nicely organised" folders, then why bother with the PSE Organizer in the first place? You have extra work (must import the photos) and more limitations (must do all photo management inside of the PSE Organizer), and its hard to see any benefit from using the PSE Organizer. You won't achieve any "better organization" than what you have now.
If your goals are to use PSE tools to organize (tags, captions notes), in addition to (or as a replacement for) your folder organization, then the PSE Organizer can be very useful.
Just to clarify:
nicely organised in folders and sub folders in Bridge CS3
They're not actually "in" Bridge. Bridge is just a browser and it just shows you the current state of any folder you point it to. This is very different from an asset management program like organizer, where you must commit to using organizer to do all file moving, deleting, etc. if you want the organizer to keep track of your images.