These guys have pretty low health once tou tame them (on Xbox, at least.) So, once you finally tame one, be sure to upgrade their health and stamina because it's pretty horrible at first. Also, you can't upgrade their speed, so I suggest taming a higher level griffin if you want a faster mount.
Get yourself a vulture and go kill ovis, but let the vulture harvest it (you can't put anything in it's inventory, it has to harvest it on its own), and the vulture will keep raw mutton fresh for 2 hours while you trap and knock out your griffin. No kibble needed, nearly max taming efficiency for the cost of a few spoiled meat to tame a vulture.