After a completing this mission a couple of times in the Grizzly tank and not getting the achievement, here's what worked for me.
There are four Mega Turrets to destroy in this mission. They are shown on your radar as large red dots. The Grizzly tank (M850 - duel cannons) is the tank you start right next to, and there is also a Scorpion tank (M808B - single cannon) that you can use.
Do not get in the Grizzly, instead get in the Scorpion and clear the map of Covenant forces, but do not destroy the Mega Turrets. If the Scorpion is about to be destroyed, get out and continue on foot. Once the map is clear, go back to the start and get in the Grizzly. If a Spartan tries to get on, get back out and kill them.
Head for the first Mega Turret and use the main cannons to destroy the Mega Turret. If you use the Machine Gun you may not get the achievement.
After you have destroyed the fourth Mega Turret you should get the achievement, you do not have to complete the mission.