This achievement sounds daunting, but it's not nearly as hard as you may think, if you have some friends to play with. Personally I consider Halo 3 Legendary to be the easiest among the 4 main Halo titles, especially when you consider that it can be done with 4 coop players.
You have 10 missions to beat in 3 hours, but one of them is a cut scene, which does NOT count against your time. So with 9 actual missions, you're looking to average 19:59 per mission to finish with a total time of 2:59:59 or less.
You can check your stats by going to Halowaypoint.com, signing in with your Microsoft/XBL account. Click on games, select Master Chief Collection. Now select service record. scroll down and you have drop down menus that allow you to see your highest completed difficulty, score, time, and whether you did it in Solo or Coop for each individual campaign. Add all of these times together and you will know how much time you need to shave.
For actual gameplay, try just walking through as much of the mission as possible. You can do this by watching speed runs and seeing what paths the players take to avoid conflict. If you have a good team of four, you can also easily wipe out a section of enemies with little effort, or even have 3 people kill enemies with one guy running through while enemies are distracted with all the bullets.
Also with coop, if one person is alive, your progress is saved and you can continue to respawn. Vehicles will likely be your biggest foe, with ghosts and Wraith turrents draining your shields rapidly. Use the EMP feature on plasma pistols, as well as mounting and beating vehicles to death. Obviously try and use Tanks, gauss hogs, Spartan Lasers, Rockets, etc., efficiently against these enemies.
Also know where objectives are, what spawns enemies, where ammo and weapon caches are, and just have a good knowledge of the mission. Playing the missions for par time on easier difficulties, or just playing for fun (shocking, I know) is a good way to just learn the game.
If you have any questions or need a tip on a particular mission or section of the game, feel free to comment or PM and ask. I have beaten the Halo 3 campaign at least 100 times since it's inception on the 360, so I have a general idea on what I'm doing. Let me feel like all that time wasn't wasted for completely no reason!
I ran with a 4 person party. We did this in one session two days after the game came out, with basically no prep just using our knowledge of the game to play quickly.We were still pretty casual about it. We only restarted one mission because we died after 8 minutes when we received no check point. Which also brings me to this note: If you hit Restart Mission from the pause menu, it resets the time counter, so no harm done if you got off to a bad start.
Here are my times:
Sierra 117 - 13:44
Crow's Nest - 13:54
Tsavo Highway - 8:15
The Storm - 13:39
Floodgate - 7:21
The Ark - 27:12
The Covenant - 33:02
Cortana - 12:36
Halo - 24:43
As you can see, The Ark, The Covenant, and Halo took about half the time, but you can see how quickly the other missions were done. Just be patient, keep trying and shave time from easier missions! Don't be afraid to take too long on one or two, just play them all, and replay the mission you feel most comfortable. Good luck!