Quick walkthrough on the panda mount (which I messed up in the first beta)
1: run through the main quest until you get to a city, there will be a fountain. If you f-key the quests rapidly it does not take long.
2: wash your hands
3: time passes (wait one day real-time)
4: rinse/repeat (yuk yuk)
I tried doing it on the last day of the beta, did not realize you could not complete it unless you do it on 3 separate days.