I would never recommend just letting them all hang out together to see what happens, and I'm sure the cat eating the canary cliche is enough to give anyone pause, but I have never had one problem with my cat and my chicks interacting. That's because she (the cat) will NOT go near the chicks inside or chickens outside. When I got my very first chick who was just 3 days old, I decided I would hold the chick and put it sort of near the cat and see what happened (knowing I could intervene if necessary). At first, it was just a staring contest.
Then, the chick peeped a few times.
That cat ran out of the room faster than I have ever see a cat run and hid under the bed for hours. Hilarious! Now, I just leave the chicks in the dog crate which is in the same room as the cat's litter box. She only goes in to use the box if it's quiet (like at night or when no one in the house is making much noise to attract the chicks' attention). Otherwise, she wants to be as far away as possible. I'm sure this is not the norm, though.
Oh, and she is an indoor/outdoor cat, so if she really has to go, she will just go slap the door a few times until I open it - I don't make her hold it! haha