Please post a picture that illustrates what you're describing.

You have to address it with the neighbor. He is sticking it to you and you are worried about upsetting him? It is not legal to create a dam so water builds up on your neighbor's property. If you do nothing it makes it easier for him to say that it was always like this.

Here is a picture. Many thanks.

Here is another

Agree with laag. The neighbor cannot impede the natural flow of water draining from your property. The longer you wait to act in doing something about this, the more difficult others may make it for you. Start with the friendly chat, but don't hesitate to go to municipal authority or a lawyer if you don't get results. Don't try to solve this by "drilling holes" or similar. It won't work as a long term solution.

If it were me I would address it with the neighbor in a friendly collaborative way.
If the board was not there the surface water would be sheet flowing down to his property.
If you both chipped in for a trench drain, assuming that the grade is working with you in a positive slope and there is a place to exit the water to then both properties would benefit greatly. win win for both property owners.
Right now it is a win - lose situation and not very neighborly.

It is completely normal for sheet flow to cross property lines. Each downhill neighbor deals with it (usually by doing nothing.) As long as one does not concentrate water and dump it on his neighbor, there shouldn't be a need to be concerned. It's not being a bad neighbor, but just allowing a natural phenomenon to occur.

Thanks for the input. I'd like to try the hole drilling because I have a strong sense this guy is gonna dig his heals in.

"...because I have a strong sense this guy is gonna dig his heels in." If you have this belief and don't protect your property rights, it's likely the neighbor will make it more difficult as time passes. 1" holes will be worthless in a heavy rain storm.

Buddy, over my career I have consulted with a number of folks whose neighbor screwed them re: water movement/drainage and they're trying to fix it while tiptoeing around Mr. Selfish. What happens is the cheap fixes (like 1" holes) fail like a coked-up high school football player in an AP exam and the expensive fixes, they're not willing to do. So they suffer, they deal with it, and eventually they try to sell and it becomes a liability for selling (god help you if it rains before a showing).

So in short, your neighbor? F*** that guy. He was being completely selfish and unreasonable to do what he did. When my neighbor behind me regraded their lot when their basement wall collapsed (should never have allowed a house there, there's an underground spring behind the wall) and I got 6" of standing water in my yard, I made two calls. One to the property owner, who told me to bite it; and one to the town, who made him fix it. Boom. Done. Problem solved.

Don't be the George McFly to his Biff.

oh and d-d is an amazing goddess of design but I'm going to disagree here. Why should you have to dig into your pocket? If the homes are part of a subdivision there was probably a site plan filed with the county that showed surface flow through the backyards to a lower drain/drainage area. He's the one who decided to subvert that. Let him fix it on his own.