In a deep medium nonreactive bowl, combine the first 9 ingredients, blending well with a wire whisk. Or, combine ingredients in a large zip-type plastic bag; seal and shake vigorously to mix. Add beef or chicken pieces, coating each well with marinade. Cover or seal and marinate in refrigerator for 1 to 2 hours.
Drain steak or chicken pieces. In a large heavy skillet, sauté beef or chicken pieces in 1 to 2 Tbsp of marinade over moderate heat, turning frequently until done, about 5 to 6 minutes total. Do not overcook. Remove beef or chicken pieces with a slotted spoon; cover and set aside.
Add onion, sweet peppers, and mushrooms to skillet and sauté over moderate heat until just tender but not soft.
Arrange tortillas on a flat surface; spoon meat, sautéed vegetables, and some chopped chili pepper onto each tortilla, dividing evenly. Top each with a spoonful of salsa if desired; roll (up) each tortilla jellyroll style. Place each filled tortilla, seam-side down, on a sheet of heavy-aluminum foil; sprinkle cheese over each, dividing evenly. Carefully close foil around tortillas, sealing well. Wraps may be eaten at room temperature or unwrapped and reheated in a microwave oven at 50 to 60% power.
Servings: