How close to your garden are the nearest trees and what kind are they? Lots of people don't realize that the most important part of a tree's root system is VERY shallow and very far reaching. These fibrous roots are extremely efficient at hogging water and soluble nutrients in the soil. Most use the word "mat " to describe these roots.

Does all of this seem like what you're experiencing? If not, I have another scenario for you. :-)

The garden isn't that close to a tree; maybe 20 feet or so. I don't think it is from the tree. In other areas of the yard where I have dug to plant roses etc, the ground has a thick layer of what looks like peat moss. When I tap on it or walk on it, it feels like a cushion of roots. This mass is 3" thick in some places. I'd be interested in your other scenario.
Thanks,
Vince

I reclaimed an area a few years ago for my garden, and it was loaded with roots in the way that you describe here. One of the most problematic was the roots of trailing bamboo, which I'm still struggling with. They're long and spiky and horrible -- but that doesn't sound like your problem. But they were one root among many.

I ended up renting a tiller from Home Depot and dug down as far as possible and broke up as many roots as possible. It was successful, but it was *a lot* of work (they kept wrapping around the tiller).

If you do that, you'll want to make raised beds and then cover the rest with newspaper and organic material (or lawn fabric). You're going to stir up a gazillion new weed seeds.