Usually, there is no way for you to determine the date of manufacture but you can make a very close guess or to refresh your memory you will need to know the revision number of the Mobo PCB.
Google for the rev# of the board and you get the approx date of release of the board.
Then depending on where you're situated you can calculate the time it hit your market.
All though thats about it, I guess you're asking the question to see if they have something like seagate has on it's website for warranty claims, right?
If thats the case an your mobo is giving you a problem, I'd suggest RMA it online through their facility you will need a few details that are present though.
Figure it out, cos I can't spell it all, online, it'll work.