However, running the application under Gnome 3 doesn't work - no icon appears anywhere. Is there a similar package that does the same thing as the
sni-qt package but for Gnome 3?
The QSystemTrayIcon you looking for is already there in GNOME 3. The icon shows up in the messaging tray (lower right hand corner) but the messaging tray itself is not visible by design.
I just snapped a picture to soothe you:
sni-qt automatically converts Qt systray icons to Unity indicators. No packages do the same thing for Gnome Shell, but you could find an extension to have the missing icon showed in the panel.