Ok, so I dug through the source and the guide book to figure out the background colour id for the PopupMenu.
Unfortunately using transparent colour values causes the menu to disappear behind the main app window. It's clearly visible on the desktop if it spawns at or near the edge of the app window. I've just tried it with the latest source and this is on Windows XP.
The attached is what I'm ultimately aiming for. I figured I could achieve this with a completely transparent background big enough to draw the rounded rect and left arrow, then nudge the position of the menu items accordingly.
PS Adding the menu windows to the desktop so they're not bound by the graphics window is a real class act -- just one of many things that sets JUCE apart. :)