Well, I figured out what the problem was. It likely affects nobody but me, but just in case:
If you use the 3rd party app, VistaSwitcher (the Vista/Windows 7 replacement to TaskSwitchXP), and you enable the "Replace the Right Mouse Button + Wheel" option, it will break right-click dragging functionality such as right-click drag-and-drop, as well as mouse gestures in Firefox. So disable it if you care about those things.
Also, I realized that middle-click drag-and-drop does the same thing as the right-click version, in case anybody cares.