Here's my long story:
I've made the same mistake with my chickens.  The neighbor next to me has an empty field about two normal sized lots between the fence and his house, which the chickens love to search through.  Luckily the one time I asked he said it was fine with them being there.  So until he says otherwise I just make sure they aren't getting too close to his house.  I got a little complacent after that conversation.
Across the road from me is a problem.  I had no idea my chickens were crossing until one time I saw them basically at my neighbor's front door.  I called them back and then laughed about it.  I didn't realize they had been doing that regularly until I got a very angry voicemail.  I had never spoken to them and was very intimidated so I didn't go talk to them then (I regret this now.)
I tried many solutions but the best one for me was bird net.  I got a 7ft tall 100ft roll and hung it across the lawn from the fence to the house.  It worked better than my fence since they couldn't fly to the top, and the bottom caught on the grass so they couldn't find a way under. 
Anyway every once in a while I had to take it down for some reason or one of the chickens was particularly clever and got out.  I didn't know it was happening, but again they were going to the neighbor's house and messing up their front garden.  After another message from them, I finally went over to talk.  They were extremely nice in person.  Basically they just asked me to keep the chickens away, but if they did get into their lawn, help them clean up any mess.  It was a very relieving conversation. 
I always wondered what was drawing them to cross the street on that side, into that lawn all the time.  Then I was talking to their next door neighbor and she mentioned she loves to toss them treats over her fence, which is straight into the neighbor's lawn!  Hilarious.  I need a sign that says, "Don't feed the chickens!" or "Feed a chicken once, and she will never leave you."
Moral of the story: Talk to your neighbor in person.  I don't know what your set-up/lawn looks like, but I personally prefer putting up a barrier rather than making a run.  One side is easier than four sides, lol.  If you post pictures of your area, I'm sure we could brainstorm a solution.