Here's the problem with that. If you have a breed that instinctively goes for birds or other small animals, it's going to happen regardless of your vigilance. Unless you take your dog out on a leash in your backyard everytime you let him out.
My dog is on a leash most of the time he goes out. He is now old and slow so he would have a hard time catching anything even in the very tiny fenced yard that we have. I'm talking square feet, not square yards. He is most often walked on a leash several times a day and is only in his yard for a few minutes a few times a week.(and even then he is supervised) So I am very vigilant. Also even though he is 162 lbs his breed is LGD group, they protect,not chase for fun/hunting.