This run was done using a Demon Hunter.

I'm not sure if rules for different classes is accepted but, I went ahead and did a run with the Demon Hunter just for the fun of it.

The Demon Hunter's mobility is both limiting and faster than the monk in different ways. It's limiting in that He has no way to "teleport" like the monk can through some walls and over pits so he has to go the long way around things like stairs. He makes up for it in just out-and-out speed with the ability to both boost his speed by 60% for 2 seconds with the Vault/Tactical Advantage combo and again by 100% with Smoke Screen with the Displacement rune.