yeah that looks like a disaster waiting to happen. Plus, the neighbor's house is on higher ground with the gutter also pointing towards your house. You don't have a gutter extender on your gutter either. That area is gonna get sooo much water damage.

Look into putting in a French drain as they aren't that hard to install--it's basically a trench with some gravel to allow the water to drain quicker into the ground. I'd also install a catch basin or two around your property since yours is lower than your neighbors. The last thing you need is water damage to your foundation. Keep an eye on your crawl space [assuming you have one] to make sure you don't end up with water under your house. If it becomes a problem, install a sump pump.

French drain impossible due to underground utilities and concrete at both ends

My question wasn't could I put dry creek bed in or not it was suggestions on what it could look like

I think what people are trying to say nicely is that it would be a bad idea to do here, unless you want it to end up being a wet creek bed with a house floating in the middle. How about just some lovely plants and garden beds that'll help soak up the water? You could do it in the meandering path it looks like you've already created.

Get a plastic culvert and cut it in half. Embed it in a trench you have dug by hand. Put it as close to the property line as possible. Fill it with river rock. Make sure it is inclined down to the road.
Lengthen you down spout so it empties into the rock.
Hope this works. Nothing is worse than water collecting by the house

They sell long flexible tubing that you can stick on your downspout and direct the water away from the house. Your neighbor needs to get one too since their water is going into your yard. How about rocks against the house with stepping stones and ground cover. Easy maintenance.

I truly appreciate everyones comments. I think that it is my fault with how I asked the question. Allow me to rephrase. There has already been a trench dug underneath the landscaping cloth that you see. I was asking for some design ideas with the dry creek bed not advice on would it work or not. Sorry that I wasn't more clear

how did your creed bed turn out?