Bring a large pot of salted water to a boil and cook the lasagna noodles until just tender. Drain them and lay them flat on a clean surface to cool slightly.
Heat a skillet over medium heat. Add the ground beef and cook until browned, breaking it up as it cooks. Stir in the grated zucchini and chopped spinach and cook for 3 minutes until the vegetables soften.
Remove the skillet from the heat. Stir in the ricotta cheese, oregano, and a pinch of salt and pepper until everything is evenly mixed.
Spread 1 cup of marinara sauce across the bottom of a 9x13 baking dish. Spoon the beef and vegetable mixture along each lasagna noodle, then roll them up and place them seam-side down in the dish.
Pour the remaining marinara sauce over the rolls and sprinkle the mozzarella and Parmesan cheese on top. Bake at 375°F for 20 minutes until the cheese is melted and bubbly.