I hate to tell you this, but if neighbors are complaining about being awakened at 3:00 AM, then in the spirit of being a good neighbor, get rid of the rooster. I live in town and have 20 hens in the backyard. Our yard is small and we have close neighbors on all sides. The hens make very little noise and nobody minds them, especially when "in the spirit of being a good neighbor" I give them a carton of eggs.
Your neighbor will go from politely asking you to hush up the rooster, to complaining to the authorities, which could result in having to get rid of ALL your chickens. Put yourself in the neighbors place, people have to get up and go to work, church, school, etc. and if they are not getting enough sleep, it affects their job performance and makes them grumpy all day. Enough grumpiness, and it will land on your doorstep.
By all means, enjoy your chickens. You don't have to have a rooster to keep a flock of hens. Peace in the neighborhood is worth more than any rooster.