I want to say Resident Evil and Dead Space made good use of environmental sounds (wind blowing, screech of a violin, some creepy echoing and/or wailing sounds), that definitely fill in the void. Most horror hlgames are about atmosphere, and a good one would use minimal music to not distract from the tension.

For non-horror action games, I can't think of any off the top my head. It would also really depend on the setting to know what environmental sounds would be available. Some games may use a chorus that is sung, but again not sure if a singing choir would be considered music to you.

But, I'm really interested in what you come up with, a game using only environmental sounds is intriguing.