Well, judging by rough estimates, it seem to me that the mini-tower designs for the Precision 340 and 360 are very, very similar, if not identical.  They seem a bit differently-configured, though, than most of the Dimension designs, so I can't say for sure if your motherboard would come out of a 340 and go into a consumer-level machine.  I'd be cautious of even trying a 360 chassis, though - I don't have one to open up in front of me right now, but I do see that the 340 has a really strangely-shaped motherboard... and the screw configuration to hold the motherboard down is definitely an oddity.  It certainly won't go into a regular ATX case without a lot of complaining and/or handiwork.