1. On waxed paper, coat riblets with flour.
2. In 5-quart Dutch oven over medium-high heat, in 2 tablespoons hot salad oil, brown riblets, a few at a time, removing them to bowl as they brown and adding more oil if needed.
3. In drippings in Dutch oven, cook onions, until browned. With slotted spoon, remove onions to medium-sized bowl. In remaining drippings, cook mushrooms 5 minutes; with slotted spoon, remove to bowl with onions.
4. Spoon fat from Dutch oven. Pour wine into Dutch oven; stir and scrape to loosen browned bits. Return riblets to Dutch oven; add tomatoes with their liquid, salt, sugar, and thyme. Over high heat, heat to boiling; reduce heat to low; cover and simmer 2 hours or until riblets are fork-tender. During last 30 minutes of cooking time, add mushrooms and onions.
5. About 30 minutes before serving, prepare rice as label directs.
6. To serve, with slotted spoon, place riblets and vegetables on large deep platter. Pour sauce over riblets.