If anyone out there has a way to train their dogs without themselves being there, please let us all know how you do it! This seems like a neat trick. I'd love to have my dog trained while I'm away.
In response to the OP the only way that you can ensure that you pets are safe and not disturbing the neighbors is to keep them crated inside. That may not be the response you wanted to hear and it's probably not the solution you're going to use. My family and I were in a similar situation as you with annoying neighbors. Our family dog used to be kept outside and as typical dogs do she would bark at the people who walked behind our house and the birds that landed in her yard. The neighbors decided to take matters into thier own hands and broke into our house & stole our dog. They returned her later that night after we got home, they had taped her mouth shut and legs together with duck tape. We were told by them that she jumped the 6ft high fence and if she did it again they would tape her up like that and throw her in the delta. They were not very nice people to say the least. We then tried keeping our dogs in the garage while we were gone for the day. That ended up with them destroying lots things, incuding chewing threw sheet metal, and making lots of escapes everytime we opened the garage door. Lastly we decided to keep them crated inside and it's been the best descision we could have ever made. There were still problems with the neighbor harassing them and us when we went outside, but we didn't have to worry if they were going to be safe during the day. The only other option that we didn't try was putting a kennel in our backyard. That was ruled out because of it's expense and the fact that the neighbors could still hear and harass our dogs.
Hopefully your new neighbors will be better than your last and you'll find an effective solution that keeps everyone happy. Who knows maybe your new neighbors have xeriscape landscaping and won't need a lawn mower at all