This one is tricky because there are several ways to destroy turrets without counting toward the achievement. To wit, turrets destroyed by environmental damage, enemy damage (via Possession), and damage over time (when the turret is at low health) will not count toward the achievement. All ten turrets must be directly destroyed by your weapons and/or plasmids.
For best results, play on easy mode and feel free to exploit the checkpoint system to destroy the same turret over and over again. You do not have to destroy all ten turrets in one play-through. The counter is cumulative across any number of play-throughs or checkpoint reloads.