Most modern Linux compositing WMs have a visual effects plugin that adds external shadow to any window including popup menus. But popup menus usually have a different kind of shadow. Simply converting all GTK+ LnF popups to heavyweight components does not improve the appearance because the standard window shadow makes popups look different than native popups. Probably, changing AWT popup window type or setting special hints to it could help.
Also, if heavyweight popups will be used in the solution, it is necessary to test if heavyweight popups are able to overlap a full-screen window on all possible Linux/Solaris versions and distributives.