There seems to be a lot of question about what counts, and what doesn't count, toward this achievement. The way the game counts even the same type of trap also seems wildly inconsistent. So here's a list of the various traps and the degree to which they seem to count toward the achievement.
Skull traps with the laser trip beam: If you destroy the skulls before actually setting off the trap, it seems to almost always count toward your progress. Occasionally just walking through the beam and setting the trap off has counted for me as well.
Trapped chests: These are the long, white chests with a light beam shining down on them. Occasionally when you open them you get some nice loot. Mostly when you open them they spawn three turrets around you. Setting these off almost always counted toward my progress on this achievement.
Nerve gas trap rooms: These are the rooms that lock when you enter and a green nerve gas spreads through the room, inflicting constant damage on you until you exit. You must shoot, or otherwise destroy one of the doors to the room to get out. These mostly always counted for me.
Alarm beacons: These are the tall, narrow poles that generate a countdown timer when you get near them. If you don't destroy the beacon before the timer expires, enemies will spawn in. Destroying the beacon before the timer expired occasionally counted toward achievement progress for me.
Mines: These are small, hard to see anti-personnel mines that you will find scattered throughout the missions. As you'd expect, they explode and damge you if you get too close to them. You can shoot them from a distance to safely destroy them. Doing so never counted for me toward achievement progress, but people have reported that it counted for them.
Stationary turrets: Not really a trap, per se, but again people have reported that destroying them counted toward achievement progress. They never counted for me.