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down vote
If you have a couple of days before the move, and a cohort at the destination, then:
Duplicate one of the hard drives (or simply use an existing duplicate) and mail it to the destination.
Once the drive arrives, have the recipient checksum the result, and compare it to a checksum of the one you have.
If they match, ship your computer as-is with no worry, knowing that you have a backup at the destination in case you need it.
Alternatively, leave one disk where you are now, and have it mailed or destroyed once you arrive and verify the data.
depends on the mode of transportation, if by car it should be fine as long as its not met with many bumps and things of the like, if by plane, i suggest packing peanuts, (by boat it would be fine), just keep it cooled, and secured in place and all should be fine
All the parts that are not mechanical (i.e. not the hard drives) can travel without being taken out of the case.
Your biggest challenge is shock and extreme conditions for the drives, so pack them tightly in anti-static bags and bubble-wrap, and possibly bind them in plastic (to create sufficient air pockets for maintaining temperature), and of course separately, though you can put them in the same box.
Hard drives travel all over the world on a daily basis, packed in a lot less than this and Dacota's suggestion, so you should be fine with a bit of common sense.
For what it's worth, I travelled through three countries with three hard drives in my carry-on luggage, each in an external chassis, wrapped in bubble-wrap. They are all still running nine months later.
down vote
Will my system be okay if I leave everything - video card and hard drives in the same box?
If they are traveling via car/truck, they should be fine. At your destination, make sure all the cards and cables are fully seated. And don't forget to change the timezone setting after booting up.
Something useful from all the office moves I've made with computers is to put all the cables associated with a particular computer in the same bag. Label the bag clearly with the computer's name.