Place the dried porcini in a bowl, cover with warm water, and set aside to soak for 1 hour.
Drain and pat dry on paper towels.
Cut any very large oyster mushrooms into 1 1/2- to 2-inch (4 to 5 cm) pieces. Cut any very large shiitake mushrooms into halves or quarters.
Heat the oil in a large, heavy skillet. Add all the mushrooms and garlic and saute over medium heat until they are lightly browned and tender, about 10 minutes. Halfway through the cooking, add the rosemary.
Season with salt and pepper, stir in the parsley, and set aside while preparing the polenta.
When the polenta is done, briefly reheat the mushroom mixture and serve hot over the hot polenta.