I know of plenty of parents that let their babies sleep on their tummies. That said, until a baby is able to lift his head and control his body a little better, it is still a bit dangerous to let them sleep on their tummies.
You will get 'yeses' and 'nos' from both sides but, ultimately, it's your decision. If you do decide to let him sleep on his tummy after 3 am, make sure that he is sleeping on a very firm mattress and that there are no blankets, pillows, toys or extra items in the crib.