This is one of those topics that tend to polarise opinions (especially by those who have not thought about this issue a bit).
S/W RAID (as implemented in Windows, Linux, etc) means you not dependent on the peculiarities of the hardware RAID controller. It is also a tiny bit slower in performance than H/W RAID.
H/W RAID means that your OS needs nothing clever or special - it does not even know its talking to a RAID volume (or at least, in principle).
H/W RAID generally means, though, that you need the exact same RAID controller in order to recover anything.
In any system like this you need to analyse the weak points: the single points of failure. For example, a drive might fail, and if you use striping then you lose the lot. If you use mirroring (or RAID 5, or a combo like RAID 1+0) then you don't. If you lose a h/w RAID controller, you need another one the same to be sure of getting stuff back (potentially you can lose the lot). If you lose a processor or RAM then you need replacements, or some means of getting the controller or drives on new hardware (or you lose the lot).
Each of these has associated risks and ease of replacement if the broken item.
As pointed out before, RAID is not a substitute for backup. Understanding all that, if you want to reduce your exposure to single point failure events, then the smallest amount of hardware "stuff" is probably the better way to go, which means, s/w RAID done by the OS and wear the performance hit.
This then means that in your case, that if you can get the exact same H/W RAID controller your chances of getting the data off are improved (but not 100% - depends on what went wrong). If you can't get the same controller, the chances go down. Possibly to zero. You won't know until you try, but don't go betting money on the outcome.
If you buy the exact same m/b, and plug all the drives in, you may get somewhere. You may not. You say the machine won't boot, and the cause of that might be that the boot disk / sector / information / partition has been hosed. If thats the case it does not matter what h/w you try and use, your chances are not good.
If the cause is the raid controller being broken you may get somewhere. It's all dependent on what caused it to break.
In the case of Rolniks question: My experiences with h/w RAID have been generally unhappy either in the setup or the long term outcome (I've seen many h/w RAID 5 systems that suffer a 1 disk failure and can't recover when they should - thats what its for!). I've used a number of s/w RAID systems (Solaris and Windows) and had no troubles at all with rebuilds, recovery, etc.
I've also picked up a 2-drive mirror from Windows, and migrated it from XP to Win 7 on the same box with no data loss. And then plugged it straight into completely new hardware and had it all "just work". Again no data loss.
For me, everything to do with S/W RAID has been a winner. (Also means just pick a mobo with lots of plain ole SATA ports on, and plug in drives... easy and cheap to set up).