From the Rhythmbox FAQ:

From where is artwork fetched?

Rhythmbox uses last.fm to look up artwork.
If you have local image files, saved in the same directory as the audio file(s) with the filename "cover", "album", "albumart", ".folder", "folder" or "$artist - $album" Rhythmbox will use them instead of looking up a cover on last.fm. It will also use images extracted from file tags.

So, the answer will depend on how your artwork is stored in iTunes. If it is embedded in the actual AAC/MP3 file, or exists on last.fm for Rhythymbox, it will show up automagically in Rhythmbox. If it isn't in one of those two places, Rhythmbox looks for specific filenames (listed in the FAQ answer above).

If your artwork is external (i.e., not embedded) you would want to include those files in your Dropbox folder, or use a third-party tool to embed the artwork.

I'm not an iTunes user, but if I recall correctly, it stores artwork in obfuscated directories and with obfuscated filenames. If this is still the case, copying the artwork won't matter because the file will be named something Rhythmbox won't be able to read.

You could try this plug-in I found (but have never used), that says it will download and embed the artwork for you:

http://www.freewaregenius.com/2007/11/11/itunes-art-importer/

Good luck!