Me, too! I first noticed these roots 7-8 years ago. At first, they were only coming in a few feet from the south side my 25' x 70' garden. Now they are all over. I've taken these to the county extension office and several garden centers. The only answer I get is, "They're tree roots". There is an oak tree (probably 200 year old) 75' away in a neighbor's yard to the east. To the west is an huge maple tree 50' away. Between my dad and I, we've had a garden in this exact spot for more than 50 years. If the roots were from those two trees, why would they have just shown up in the past few years?
I have to till every spring and still play tug-of-war digging my dahlia tubers in the fall. I'm wondering if they might be from something in potting mix or top soil.