Yes, you can start new walnuts from the nuts your trees drop. Simply collect them in the fall and bury them under a couple of inches of soil and/or mulch. The nuts need to experience a winter outside to germinate. The roots of walnut trees give off a substance called juglone that inhibits the growth of other plants and is toxic to some species such as tomatoes. It's a natural way the trees fight off competitors. That's why growing veggies and certain other plants is very hard to do and not recommended."