What would you do?

Can you lock your gate? Like with a bike lock maybe?

Get a lock for the gate & get a video cam set up to see exactly what's happening. If it's your neighbor, & you catch them on video, you can either confront them, or take your proof to the police. At the very least they are trespassing.

I like the video camera idea, you could try that

Honestly, I think you should secure your dog in a safe place, free from the elements. Poor thing sits in the sun or rain all day? Aside from that, I had a neighbor who let his dog out all hours of the night and day. It barked and barked the entire time! I would just put DD down for a nap, and the dog would wake her up all the time. My DH worked nights at that time, and he couldn't sleep during the day because the dog would bark. I couldn't sleep at night, because they'd let it out at 2am! If someone trespassed on your property, and let your dog loose, then they are very wrong/broke the law. At the same time, I can sympathize with your neighbors. If you're not at home, you don't know how disruptive your dog is being.

maybe No Trespassing & Beware of Dog signs? or you could get a crate & keep the dog inside while you're gone.

Go to all your neighbors and just tell them. "Just in case you are the one who is letting my dog loose. Stop it!"

I really think you need to be more considerate of your neighbors. You don't know what their schedule entails, and you don't know how loud/whiney/bark-a-holic your dog is actually being. I would set a camera up just to video my dog and see how disturbing his behavior really is, then I might leave a letter on each of my neighbor's doors apologizing for the annoyance and assuring them that I will figure out a better solution.

Your lucky. If I were your neighbor and your dog is obviously the annoyance you didn't think he was, I would probably bang on some pots and pans in the middle of the night outside your window just to disturb your night just like your dog disturbed my day.