Do a Barrel Roll!

Actually you just have to press Z or R twice to get this achievement.

Since Wowhead is only showing "Alysrazor slain" 4 times...

Here is the ingame criteria aka stuff you should not get hit by (as of patch 4.2):

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• Brushfire (Flying Phase): The Blazing Talon Initiate conjures a fiery ball that moves across the arena.
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• Incendiary Cloud (Flying Phase): Alysrazor periodically creates an increasing number of Incendiary Clouds lasting for 3 seconds.
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• Lava Spew (Flying Phase): Plump Lava Worms spew a molten cone of fire, dealing damage to all enemies within a 35 degree cone.
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• Fiery Tornado (Transition after Flying Phase): Fiery Tornadoes erupt from the Fiery Vortex and begin moving rapidly around Alysrazor's arena in circles.
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Remember: The achievement says, every raidmember has to avoid everything.

This achievement looks like it's based on the Mimiron achievement Set Up Us the Bomb, which could be done in multiple parts, meaning that one week your raiders could focus on not getting hit by Brushfires, Clouds and Lava Spews, and the next week you strictly avoid the Vortexes.
Also the achievement criteria counts for each raidmember. So following the example above, if someone is not able to raid the second week, where you avoid the Vortexes, he still has completed the 3 other criteria and so it is possible for him to get another raid in which every raidmember avoids the Vortexes and so he still gets his achievement.

Long story short: Every raidmember has to avoid things, but each raidmember gets personal credit.

Edit: Ascendance on EU-Alonsus was the first guild to complete this achievement, so they also were the first ones who completed Glory of the Firelands Raider.
Other Edits: Typos ;)

This is one of the popular quotes said by Peppy in Star Fox 64. It's pretty nice for Blizzard to put in a bunny icon as a tribute to this quote.