Telling the truth
Unless you've had nuisance dog problems before, you won't be aware of it straight away. You'll hear barking and you'll think it's some stray that'll soon go away. Then you hear it again and wonder where it's coming from but you won't be too bothered. It probably takes a couple of weeks to realise that you've got a serious noise pollution problem. Once you realise what's going on, you basically have two options.
Option 1: This is what most people do. They try to ignore the dog as long as they can. They will rehearse the conversation with the neighbour, but do nothing. Eventually, they summon up the courage to confront the neighbour who will be at best indifferent, at worst hostile or even violent. The carefully planned conversation goes out of the window - the polite "sorry to be a pain, but your dog barks a lot at night" doesn't go down too well with an owner who knows all too well that his mutt is a public nuisance. "My dog doesn't bark, you liar!" "Yeah, yeah, chill man, it's only a dog!" Or "Get off my property or I'll smack your face."
Your bluff has been called. You can try writing polite letters but it will do no good. Now, if you're a millionaire and you can afford the best lawyers, you can sue the selfish dog owners. Make a point of saying that you don't want money, just quiet. Unfortunately, though, most people can't afford to do that. So it's up to the police, local authorities and whatnot who are almost always useless. If you do ever get a result, it'll be after months or even years of sleepless nights.
Ignore the comments on here that say "write a letter" or "talk to the neighbour" and such like. Neville Chamberlain tried that with Hitler and look where it got him!
Option 2 is the only sensible one. Kill the dog.
I won't repeat all the tips (some better then others) here. Just the most important points. If you have been kept awake for more than 3 nights in a week by barking, you have a problem. This is the critical point: SAY NOTHING. Don't complain. Forget polite letters or well thought out requests for cooperation because THEY WILL NOT WORK. Instead, make sure you show no annoyance at being kept up all night and if you can, pretend that you actually like the nuisance dog. Then, when opportunity presents itself, kill the dog.
Remember, you are simply eliminating vermin. If your chosen method of killing the dog is slow and painful - that's fine. But avoid going out of your way to take it out on the dog as that will get you caught.
Bottom line: Dog barks; you realise the problem; you say nothing and kill the dog.
Good luck!