Greyhounds naturally chase almost anything, well most greyhounds at least. I have a friend who owns two that have never been raced or trained, and he says that he can't let them off-leash at the local dog park because if they see a squirrel they'll be in the next county before he catches up!
Racing greyhounds are trained to focus on the lure and not get distracted by the other dogs, but normally in modern times the training isn't cruel. There have been historical cases of cruelty, and until recently retired racers weren't treated well which is where the sport got a bad name from. It is something the sport is trying to stamp out.