cd /usr/share/icons/<your_theme>/
sudo cp -v /home/<your_user>/.dropbox-dist/icons/hicolor/16x16/status/* status/22/
sudo mv icon-theme.cache icon-theme.cache.old
Replace
<your_user> and
<your_theme> as appropriated.
You may want reboot to get it working (although it worked for me after a while).
That worked on system tray panel of 12.04 LTS Precise Pangolin 64bit. I've
installed Dropbox from command-line.
This solution is based on
http://forums.dropbox.com/topic.php?id=35693.