The Root of the Behavior
If your dog is from a breed which classes him as a working dog, even though he may be just a house pet, he will have strong inherent traits which are likely to arise when he sees that brown feathered bundled legging it to safety from under his nose. Sheepdogs and even the short-legged Corgi are both dogs renowned for their excellent herding skills. If your dog is one of those or any other working breed, he will not enjoy the fact that the chickens are running around loose. In fact, he will feel inclined to round them up and get them back to where they should be because he sees it as his job. Some dogs are born to run and, over the centuries, that canine skill has been useful in helping to catch swift moving vermin like rats and rabbits. A dog, if it has not been trained to be around livestock or wildlife of any kind, does not know the difference between a chicken and a rat, so will set chase at the slightest indication something may be escaping him. It is a natural instinct which may have, as in Terrier breeds, been intensified over various generations by breeding. If you've got a Whippet or a Greyhound, you should make sure you and the chickens have got your running shoes on as chasing to them is more than second nature, it's what makes them tick. Dogs are predatory beings. If they still lived in the wild rather than in a comfortable house where the dinner is served on a daily basis by their personal assistant, aka you, they would need to hunt for their food. There is only one way for a wild dog to catch its prey and that is by chasing it. Dogs chase chickens because, even though in this day and age they shouldn't, it is a natural thing for them to do.
Encouraging the Behavior
Although your dog might enjoy chasing chickens and even get some valuable exercise from sprinting after poultry on the run, it's really not a good thing to let him do. The chicken, as it scurries across the yard squawking, may appear to be having fun as it tries to take flight, but it really isn't enjoying the moment anywhere near as much as your pup is. Dogs can cause harm to the chicken when they're chasing it, which may be completely unintentional. Although chickens appear quite sturdy creatures, your dog, by chasing a chicken, can raise its stress levels so high it will suffer a heart attack and that's without him even catching up with it or touching it. A sad fact which will not endear you or your dog to the farmer or person who owns the chicken and could possibly land you both in some proverbial hot water. If you keep chickens yourself and let them roam around free-range, your dog's instinct for chasing them could be put to good use. When dusk falls he'll probably be more than happy to round them up and assist in shooing them into their coop where they'll be safe for the night from any dog-like prowling predators such as foxes.
Other Solutions and Considerations
For a dog to chase a chicken is a big no-no. It's bad for the chicken and will ultimately upset the chicken farmer, which could cause you serious problems and even lead to a lawsuit. Let's not forget the fall in egg yield which could possibly occur if the chicken has had the fright of its life. Breakfast just somehow wouldn't be the same without easy overs or scrambled, so if in doubt, keep your pet away from chickens by keeping him on his lead.