Hi M.
It is true, your neighbors are probably used to having things a bit more quiet. Often times people have difficulty with the lifestyles of others when they are no longer sharing that lifestyle. People quickly forget what it was like to be a parent of an infant, a school age child or a teenager. All of these age groups come with different experiences.
As a mother of four ages 29-7 I have been in all these groups many times. Even now, although I've been there and I sympathize, when we go out to dinner in a restaurant, I am somewhat disturbed if someone has an infant and allows them to cry incessantly throughout the meal.
You did not say how old your dog is, but if you've had him a long time, I'm sure you don't want to get rid of him. The best thing I can suggest, although your life is likely quite stressful and you probably are feeling less than neighborly at this point. I would try to talk with your neighbors. Allow them to get to know you and your children. When they can identify and empathize with you on a personal level, they will be less likely to complain.
You could try approaching them and apologize for the barking. Ask them how much barking from a dog they consider ok and what time of day it bothers them most. Assure them you will try to keep the barking at a minimum and ask them to please call you when the barking seems excessive. Try to appear sincere in your concern for their quiet enjoyment.
Do you know what triggers your dog barking? For me my dog barks only when others walk a dog past our property. This is normal dog behavior and would be very difficult to stop. If your dog is barking more frequently than that then his barking could be a behavior in response to frustration. If you can find the cause you could hopefully limit the barking. With 3 small children you are probably very busy. Is your dog getting walked less frequently than usual? Is he lonely and understimulated in the yard? Is he hungry or thirsty, hot or cold?
Best Wishes
J. L.