Fine fibrous roots throughout my Garden, How to control or get rid of?

[A Dog Named Boo!]

I am sure they have spread from my neighbours poplar trees. I have tilled every spring, but they still survive.
So, I can grow some above ground veggies, like peas and beans, but any root crops do not do well at all. Is there any chemical or procedure I can use to get rid of those root thingies for good?
Any advice is appreciated