I hate to be the guy whose answer is "throw money at it", but a 3D mouse is a huge help with Altium (and any other 3D applications). 3DConnexion's SpaceMouse series starts at like $125 and is top notch. I really like the wireless one, because it also works wired, so you can leave the wireless receiver in your laptop and plug it in at your desk.
(In particular, you might notice that during certain 3D Body commands in Altium the Shift+Right Drag tool no longer works. Combined with Altium's hideously bad handling of 3D snap selection (actually the 2D snap selection isn't anything to write home about either), this makes the process of getting complex 3D bodies lined up properly downright torturous. A 3D Mouse continues to work during these commands, and renders the process merely painful.)