You're probably looking at a long-term project. If you don't want to use a lot of nasty poisons, you just have to dig, dig, and dig some more. Any plants that you can't dig up at a given time, make sure to remove any flowers and seedheads. Once on the ground, those seeds can pop up years later.
If it's any consolation to you, the appearance of nettles is a sign that you have good quality soil. And if all else fails, consider growing nettles as a crop - most nettles have excellent medicinal uses, and are much in demand. You could probably find a market just for the seeds.