Most of the profile data is based on text files anyway, the only issue would be the history sqlite database, which might not be Linux compatible, though I would expect it to be.
Update: I dug a little bit deeper, and found out that there will be discrepancies regarding the following profile items:
compatibility.ini : not so serious, but some compatibility information will be lost. Can be easily modified to adjust directory/path names.
extensions.ini : Because you have absolute paths to your extensions in this file, you will probably lose track of all your extensions/addons if you carry this 1/1 over to your linux user directory.
Nothing that can't be fixed with a simple search&replace, though.
prefs.js: There are a couple of entries here like browser.download.dir, browser.download.lastDir or dwhelper.storagedirectory which can be re-set easily once on linux.
search.json: Almost all search plugin meta contains absolute path names. Again, nothing disastrous. Like the extensions.ini, you can S&R.
The rest of it are either binary files or sqlite databases which can safely be transported to your linux dir, or meta text files that contain no OS-specific detail.