I have a rather large garage (30' x 30') with a concrete slab floor. I want to build a laundry room and bathroom in one corner of the garage. Are there problems I may encounter in doing this?

Without sounding like a smart-a**, I think the reason you haven't received any responses to this question is that it's not specific enough. There can be 100 answers to this one.
Of course you're going to encounter problems! Good grief... just to start, you've got a solid slab that's going to need plumbing run through it. And building plans and permits and sewer/septic tie-ins and... as stated earlier, Good Grief! LOL
Best of luck with your project!

Ok, upon further review and before anyone feels the need to jump all over my reply: Yes, theoretically you could build this project without cutting the slab.

Well... from my experience... you can pretty much ALWAYS expect a problem (or problems) when you make significant changes like adding an addition that includes pretty much all of the utilities that a bathroom/laundry room involves. There isn't really a room in a house that uses more "infrastructure" than a bath. You have electrical, plumbing and if you do it right... ventilation. AltaVista is right, too. You do need to consider your local building codes, current waste system, etc. This isn't a question that can be answered without more information. Except to say... "Yep...you'll encounter problems". The number of which depends on how much homework you do before you start.