If you use a crossbow to tame these beasts, PACE YOUR SHOTS! The last two I tamed came within one arrow of death when they finally fell. The best way to tame them is to go to the desert and grab all the red crates, hoping for an ascendant longneck. Also, my perfect (semi)Universal trap is 5x3 foundations and ceilings, stone Dino gates on the 3 wide end, and thirty pillars, three high in the center of each border foundation on the long side.
Before you all go naming your griffins "Buckbeak" lemme lay some mythology on ya. Buckbeak from Harry Potter was actually a hippogriff which is a horse combined with a eagle. A griffin is a lion combined with an eagle. Just keep this in mind before going on a Buckbeak naming spree.