In 15w50a, if you equip any armor piece by holding it and right-clicking, you'll hear a different sound according to its material. However, that is the only way you can hear them - they don't play if you equip the armor through the inventory or using a dispenser.

This only isn't a crazy idea for a simple reason: horses already produce a sound when you use their inventory to equip saddles or armor. So, I don't see why it couldn't happen with the players as well, as we've seen something similar before with the horses.