Then I fill it with logs up to the original soil line.
Then I use the excavated soil to build a berm over the log filled trench. Because I'm fighting sandy soil and Bermuda grass, I then over the whole berm with weed barrier.
I've laid down scrap lumber to make a walkway. Then I cut holes in the weed barrier and plant in the holes. These two berms have been planted with potatoes. The logs soak up water and hold it in the root zone, containing runoff. Because the logs are buried deep enough to get little oxygen they decompose in a way that does not rob nitrogen from the soil.
Eventually the logs will break down, giving back to the soil all the good nutrients the tree stored up in its lifetime. While my beds aren't to that point yet, I've already seen a big difference in how often I have to water. I water less because the rainfall stays in my beds, held by the soggy logs.