Only if the controller is the same. So if you old motherboard has Intel ICH7R chipset and your new motherboard has Intel ICH10R, then it should work. But not cross-brands. This can be done manually but is unsupported and very dangerous.
^ if you've got an extra card slot, that's definitely the easiest approach and it's pretty much guaranteed to work. Plus a dedicated RAID card might yield better performance in some cases than an onboard RAID controller.
If your dedicated card is PCI (which i suspect), it will be slower than the onboard RAID. But hey, you can check if it works, backup the data and re-create on the onboard RAID controller if you want. The speed difference may be noticeable or not, so its up to you if its worth the trouble of rebuilding the RAID on the onboard controller.
If you want to keep your data, do not write to the devices or make any changes, but perform manual recovery of RAID arrays.
This also is a good reason to run software RAID instead; but it doesn't explain how one disk lost its configuration. Are you sure nothing happened to it? What happens if you put the RAID card + HDDs back again?
So the issue now is you want to get your data back? Or you just want to setup a RAID again and don't bother about the current data?
If you want to perform recovery, you can boot from windows with the other HDDs attached to a non-RAID controller. You can then perform recovery with special software that recognises RAID; there are many search in this forum or google.
Since i don't use Windows, i may not be the best to ask. I can perform this kind of recovery under BSD or Linux, but only if the NTFS filesystem is undamaged. Unfortunately Microsoft' filesystem support is very poor (FAT and NTFS) and the successor, WinFS, was quickly dropped in the garbage bin because they couldn't figure it out correctly.
This is kind of off-topic i hope its allowed, but Microsoft couldn't even get their core product right; they had to hire DEC/Alpha engineers to help create their stable NT kernel, of which all releases since Windows 2000 (and earlier NT4..) are based on. The Win95/98/ME-kernel was, as you may know, not very stable.
Not that i disaprove of people using Windows, but especially for storage there are alot of technologies like ZFS which you don't have access to if you only run Windows. ZFS would for example be better resistent against the problem you just faced. Running Windows + NTFS meta-data only journaling + JMicron RAID is quite a fragile setup in my opinion. If you want to read about ZFS, checkout the wikipedia article here.