Pest Problems

Technically, you don't have to click on each marmot to stomp them .
My warrior heroic leaped into a group of 6 marmots and aoe stomped them.
Looks like any AOE damage will give you credit.

Not only do AoE abilities kill the Marmots as posted above, but AoEs can also kill the Insect Swarms. Just be careful not to walk through any dead Insect Swarms because they'll still slow you down.

Here's what I recommend doing for each class to make life easier when doing this daily as well:

Death Knight: Death and Decay and either [spell=48721] or [spell=49184]
Druid: Use [spell=16914], [spell=88747], and/or [spell=106785].
Hunter: Just drop an [spell=13813] on the cart where the Hozen is at and proceed to afk for 24-30 seconds. Rinse and repeat as needed.
Mage: As mentioned above, use Arcane Explosion with the glyph.
Monk: Spam Spinning Crane Kick relentlessly.
Paladin: Use Consecration or whack/judge marmots with [spell=20154] up.
Priest: The easiest way is to glyph [spell=132157] and use that. Otherwise just use [spell=124469].
Rogue: As mentioned above, spam [spell=51723].
Shaman: Drop a [spell=8190]. Supplement with other spells.
Warrior: Use [spell=6544] (preferably glyphed to increase usage) and an AoE talent of your choice (preferably [spell=46968])
Warlock: Use either [spell=1949] or [spell=5740]. Using the [spell=108508] talent is recommended.

Whenever i do this quest, I always think of This