I guess the plain simple answer is "yes, yes they will.". Cats, like dogs, are predators. They kill stuff. That's how they survived for all those years before they had humans to cater to their every whim. Unfortunately, birds can be delicate. Cats have these claws and teeth that are basically like big sharp knives with lots of bacteria on the ends of them. When they grab a bird, it's like being stabbed four or five times, or however many, and the stab wound is also inoculated with lots of really bad bacteria. It's hard for a bird to survive that. Even people sometimes get bad infections from such wounds.
I've had cats that would go out and kill full-grown cottontail rabbits. Cats also sometimes end up being the death of some creature that initially got away, but succumbed to infection. So, yes - cats can kill chickens.
This doesn't mean that they ALWAYS will. Or that they ALL do. The only reason my current cat doesn't kill things is because I don't let her outside. This is also keeping HER from getting killed by coyotes and dogs.