Repeating what Mark Henderson said on Server Fault:
If you have a dedicated RAID controller that plugs into a PCI port, then you should be fine. All of the RAID data will be stored on the controller, with matching meta-data on the drives. Then you can just move the whole thing into another server. Some controllers will even let you shuffle the drives around so that they don't need to go back in the same order that they came out in (particularly useful when you have 14 disks).
If you are using software-based RAID (i.e. in Windows or Linux), then this too can be transported between machines. With Windows, when you put all the new disks in, it will ask you to import them and they should just start running without a hitch. With Linux I don't know the procedure but I suspect it would be something similar.
If you are using an on-board RAID controller, this is where things can get tricky. You have specified that you will be moving between different hardware, so if you were moving from say an Adaptec RAID controller to a 3Ware controller, then the chances of survival are minimal. If both the boards have the same brand of controller, they may be able to read the meta-data off the disks and re-create the array.
If you're VERY brave, you can create a new array on the new controller, and make sure that you use the exact same settings as the previous controller used (same stripe size, etc), and when it asks you if you want to initialise the array, say no, and hope for the best. I've had this work with a RAID0 and a RAID10, but never with a RAID5.
So the short answer is - if you want to be able to move it around easily, invest a hundred bucks into a proper RAID controller and just move the whole thing over in one hit.
I personally use hardware RAID and have not had to do this, but I have been considering moving my 2x2TB hard drives to NAS for the household to use. Even with what they said, I would still personally take a backup if possible of at least your most important files just in case. Its always better to be safe than sorry.
To answer the question of what happens if the controller fails and you want to move the hard drives over to the new controller. You need to import these foreign hard drives into the new system. RAID stores the configuration on both the disks and the controller.
The exact method of doing this will be down to a per controller basis, but it appears you want the hard drive in the same slot numbers they were in, and you want to have a data backup on standby in case it goes awry. If you can get an identical controller, this seems to help.
It is important to remember with RAID you want a backup to go along side it. RAID keeps you going in the event of a drive failure. Backups protect you when it goes catastrophically wrong (Or you need to recover that file you accidentally deleted.)