WoWDB

I uploaded a screenshot that shows my method for achieving this. It's mostly the same as the video linked by velde046, but from the north side instead of the south side. It may be slightly easier since there were only three NPCs for me to keep track of instead of four (well, maybe three and a half, read on to understand).

Use raid markers to keep track of these NPCs - with all the other trainers running around and the many spinning blades, it's easy for the eyes to become fatigued and miss the important NPCs. I also put down both Rune of Power as a boundary marker so I would not accidentally wander out of the area and lose my progress. I didn't use smoke flares because they would obstruct my view.

Take a couple minutes to become familiar with the pattern of the marked NPCs. Every once in a while, the timing will be such that two (or all three if unlucky) will be bearing down on your position. The white path is the approximate emergency path that I would take when this would occur. When you use that path, you'll need to keep an eye out for the trainer that patrols that row. It's important to get back to your regular position as soon as possible!

Also, be comfortable with strafing movement (moving sideways). It allows you to react quicker.

Please note: The criterion will stay grey until the quest is turned in, thus you will not be awarded Ready For Raiding III (if you have the other criterion already) until you turn in the quest.

For this quest and it's companion Survival Ring: Flame, it's best to zoom out really far and have a top down view on the ring. This makes navigating all the much better.

This one's the hard one, now that it's no longer possible to cheat. If you are attempting to get the achievement Ready For Raiding III for some reason it goes wrong and you get hit, abandon the quest and get it again. Reentering the ring and resetting the timer does NOT work.

*EDIT* Removed my description and image, since there's a video on YouTube that explains it way better than I could. This is the best method really...

As of patch 5.1, the Shado-Pan Trainers now run much more closely to the pillars. (As such, the pillar trick no longer works.)