Ok so if you're here your in for the long run cause I gotta say this is a very grindy achievement.
My first advice would be to go through the campaign have fun and whatnot collecting as many relics and runes as you can because my method involves having a strong character. Play on insane and put your game on Public so you can have randoms jump in or some good friends if you have any who have the game. If your by yourself and don't feel your strong enough, Jump in an online game on insane and within an hour or two you'll be ready.
Anyway I spent about 25 hours having a blast playing with randoms, Getting stronger, Looking for the secrets (Pain in the ass btw) and trying to get these Orbs as quick as I could.
It felt like it would take forever until I jumped into a Co-op on a Trial. Basically these guys raped it in like 5 mins, So I had 27 Orbs a lot faster than I was getting.
Now here's the way I found to get 27 in TWO minutes!!!
This is my setup
Skills - Life Mastery, Overheal, Call for Medic, Touch of Healer, Shadow Fury, Like the wind, Healing Flame, Chi Mastery, Serene Mind, All 8 Elemental skills maxed, Karma Harvester. Any points you have after upgrading these put on whatever you like.
Skills are dropped randomly and given through completing certain missions!!
Armour slots - Crit damage/chance (same gem) All 3 slots
Amulet slot - Ninja touch (Melee 30% Damage)
Power slots - ALL THREE SHADOW FURY DAMAGE/DURATION VERY IMPORTANT!!!!
Multiplayer slot - Co-op player damage
Now your weapon
Blade of Exile - First slot most powerful Electric rune you have (Mine is Lvl98 Spirit of juice)
Second and third slot - Crit Damage/chance same gem.
I used a second sword for immune enemies to electric which is "Ryuken" set the same but with Toxic instead.
Using this setup my DPS is 850 and I melt everything but once Shadow fury is activated It becomes a joke as in 1350 DPS joke LOLOL.
Now go up to the Trial area and choose "Trial of Purification II" and kill a few enemies to get your Shadow Fury full and just unleash beastmode on those Mofo's. The thing about this map is there isn't any lava and there is plenty of space to run incase you get overwhelmed. Plus once you have some people in with you it will go much quicker. I was at 3000 odd Orbs before doing this method and 2 days later I got the achievement.
Good luck and I hope this helps someone get this grindfest of an achievement
UPDATE- FOXDVD has shared a very good trick below in the comments!
UPDATE 2- FOXDVD'S trick no longer works:(