What can I do to keep my teeth clean without using toothpaste or mouthwash?

Why. You should weigh the pros, cons, risks and benefits of toothpaste, mouthwash... These items have been in history for a long time, and unless you're super allergic to all of them, there's a reason why they're still used today and recommended by american dental assoc.
Brush and floss. Toothpaste helps, but brushing and flossing are key.
It not the products. You don't need toothpaste or mouthwash to clean your teeth. You need a brush and floss. That's it! a dry brush will gently and effectively remove the plaque from your teeth. Do small circles at the gumline with a soft bristle brush. Next clean the sides of the teeth with the floss. Be sure to get under the gum with the floss. The paste is for flavor and fluoride. It's not required.
Try baking soda. Try brushing your teeth with baking soda. However toothpaste and mouth wash are the best options.