By pure coincidence when looking for something completely different (not even Java related) I found this page. It is an excerpt from O'Reilly's Swing Hacks book. It contains a piece of code that gives you the effect of a seemingly translucent window by taking a snapshot of the screen contents behind it and use part of it as the background image for the dialog to be opened. This effectively makes the background shine through the parts of the dialog you do not explictly use for something else. The only thing that does not work is the automatic update in case the dialog is moved around on the screen (resizing is handled, however). But this is no problem for us, as we only use undecorated popups that cannot be moved anyways.
I wonder if I should buy the book and see what other cool stuff can be done without resorting to JNI :-)
Update: I have written a follow-up more recently. Be sure to have a look at it, too.