I've never done this, so take my advice with a grain of salt, but...
I would recommend pulling the GPU, PSU, and HDD out of the case at a minimum, and the CPU cooler if it is large/tall/heavy
The PSU just makes sense because it is quite heavy, and in some cases, only held in place by the four screws in the back. IF your PSU has a lot of support that keeps it from torquing the back panel, you may be ok leaving it in place.
The GPU, because it's expensive, but also because the shape of it, and the way it's secured, it is easily torqued back and forth inside the PCIE slot by it's own momentum if shaken. (The same goes for any other PCI cards)
The CPU cooler, because a tall CPU cooler would be similar to the GPU, it's momentum when shaken back and forth would cause it to strain the mobo/mounts. If you've got a short CPU cooler (like the stock intels) you may not have to worry so much. If you remove it, be sure to keep thermal paste on hand to stick it back on.
The HDD simply because you don't want to lose your data if the UPS guy attempts a barrel roll. I would go as far as putting it in a static-bag and putting it in my carry on (if you've got important data, though I'm not sure how airport security works, have never actually flown)
I would leave the mobo/ram/cpu in place, they are pretty secure where they are, and seem more likely to be damaged if you took them out. You may want to reseat the ram when you get there.
Optical drives are cheap enough, it doesn't really matter
