We've been to downtown Chicago a few minutes ago and I thought I'd make some recommendations for everyone who hasn't arrived here yet.
I highly recommend to NOT go by car due to a lot of traffic jams, and moreover don't get a cab since it's very expensive (about $50 to get to downtown Chicago).
The best way to travel is to go by train. It's only two blocks away from the Hyatt. At the entrace go right, at the first block go right again. Just after the highway go right again, there's the station (approx 5-10 minutes walk).
Over there you'll find the blue line train. Just take direction of downtown there. If you want to go to downtown just travel until the 'loop' (you'll see what I mean in the train). It's about a 30 minute trip and it only costs $1.75 (for that price you can go anywhere until you leave the station).
We did that last night... the bus system downtown is a bit hard to understand at first... after about 15 blocks of walking we figured out where to go (mind you it was 2:00am when we were doing this last night). They had a visitor pass box there at the station near the hotel but it was broken unfortunately.