Cook the tortellini al dente according to the package instructions. One minute before the pasta is done, add the broccoli and mushrooms. Cook for the remaining minute until the broccoli turns bright green and crisp-tender and the pasta is al dente. Drain, rinse under cool water, and drain again.
Toss the pasta mixture, the artichoke hearts, red bell peppers, dressing, and oregano in a large bowl, mixing well. Add the feta or Parmesan cheese, if desired, and toss again.
Cover and refrigerate for a minimum of 2 hours prior to serving.