With the advent of Firefox 3, the backend received an upgrade and all "place" information is now stored in a SQLite database.

~/.mozilla/firefox/[profile_name]/places.sqlite

That's just a regular sqlite3 database that you can open up and poke around in. Look in the moz_bookmarks table for a list of bookmarks, or just move that places.sqlite file to your new machine and your bookmarks will pop right up.

Update: Mozilla.org has instructions