Are you booting off the raid volume? If not, then yes, you can just transplant the RAID card and drives, and as long as the drivers are loaded properly on the new system, it should just show up.
As it was said yes as long as you are moving the controller it should be as simple as moving drives/controller. Of course an OS like linux will be painless (likely no change at all) but windows could need to have a repair done on it if the new hardware is too different (and causes a BSOD at boot).
Onboard? You have to know which chipset you were on, what you are going to, and see.
Add-in raid? Just move the raid card. In windows, you just need to set the BIOS correctly (if you want to boot from it) and then there's a good chance you'll reactivate windows.
Software raid? This normally just means using the same OS.