Like many 3s junglers, Diana struggles slightly at the beginning but improves as time goes on. Her primary issue is that she has no sustain unless you (often foolishly) invest in spell vamp. I still like her for the role, though, because her itemization is very versatile and you can adjust for unforeseen circumstances; plus, she's on the beefier side for a mage.
I suggest taking her bubble first and stacking it--helps massively with survivability. You'll want some health potions early; I normally start with Kage's plus (after waiting for 10 gp) two potions. My masteries for most 3s AP junglers are 21/9, putting points into attack speed instead of CDR and the ninth point for Defense into Perseverance. My runes are flat AS, flat armor, scaling magres, move speed (quints probably aren't ideal). Make sure you get Rylai's for the slow; beyond that, adjust your build to the situation, and definitely consider Malady + Nashor's Tooth if you don't need more burst.
I don't know Udyr as well, but in my (brief) experience it's best to take a point in Phoenix Stance at first level and then ignore it until you're forced to buy it again. Turtle Stance is important if you don't get an early Scepter (which I wouldn't), and most of your damage will of course come from Tiger Stance. If spreading out your points annoys you, I suggest stacking Tiger Stance early and buying a Vampiric Scepter plus a health potion at the start of the game instead.
Udyr of course always loves Trinity Force, but aside from that it's really situational, since Udyr tends to build some beefiness. Try Blade of the Ruined King versus people with high hp and/or who stack black cleavers and consider getting a (much improved) Spirit Visage if your opponents are AP-heavy. Boots are also an interesting choice, but I'd probably stick to Treads or Berserker's in most situations.