Heat the oven to 400°F. Shred the zucchini by rubbing it across the largest holes of a grater to measure about 1 cup. In a medium bowl, mix the pasta sauce and zucchini.
Drain the thawed spinach in a strainer, then squeeze out the excess moisture from the spinach, using paper towels or a clean kitchen towel, until the spinach is dry.
In another medium bowl, mix the spinach, cottage cheese, Parmesan cheese and oregano. Drain the mushrooms in a strainer. In an ungreased 8-inch square glass baking dish, spread 1/2 cup of the sauce mixture.
Top sauce mixture in dish with 2 noodles, placing them so they do not overlap or touch the sides of the dish because they will expand as they bake. Spread 1/4 of the remaining sauce mixture (about 1/2 cup) over the noodles.
Drop 1/4 of the spinach mixture by small spoonfuls over the sauce mixture; spread carefully, pulling with the tines of a fork if necessary. Sprinkle with 1/4 of the mushrooms and 1/2 cup of the mozzarella cheese.
Repeat layering three more times, beginning with 2 more noodles and following directions in steps 4 and 5.
Cover the dish with foil. Bake 35 minutes. Carefully remove the foil, and continue baking about 10 minutes longer or until lasagna is bubbly around the edges. Remove from the oven and let stand 10 minutes so the lasagna will become easier to cut and serve.