I've used humane live traps to catch unwanted critters (including dogs) trespassing on my property. When I catch a stray dog, I do one of two things. I turn it over to the local animal control authority (usually SPCA), or I take it 50 miles out of town and turn it lose. Either way, no more problems.

Of course, you have to be discrete about the capture and transport, but it isn't that difficult.

I guess I could just catch them and kill them, but it ain't their fault. They just need to find a better home to live in.

A live trap big enough to catch most dogs with costs me maybe $50, and I can re-use it many times, or rent it out to other like minded individuals. It is a worthwhile investment, and no one, including the dog, gets hurt that way.