Yeah you need the shroud, not the sides. But a clean radiator is a must too. They dont have to be fancy, just need to extend far enough to be around the sides of the fan and force air where you want it.
Just use a pressure washer, but be gentle with it. A cheap electric one is ideal, with a spread out nozzle, not the jet stream one. If you don't have one, just head up to the local car wash, that should do it. Doesn't need much, just enough to push the dirt out of it