To begin with, I want to say that this is my first guide so don't dislike it instantly :)
I just want to show you MY way to be a very good jungler. I use this build all the time and IMO it is very good. Hope you enjoy :)

For those, I choose Smite and Ghost.

My choice:

Smite is of course for effective jungling.
Ang I use Ghost as a second one cause it is very useful: you can escape quickly, run to defend a lane quickly, get to the teamfight quickly. Mainly on the low levels, when you still don't have your Treads and your movement speed is very low it just helps you a lot. And despite the fact that you have your Blazing Stampede which gives you a lot of movement speed, it could not be enough, for example: when champions like are chasing you. It is just the best choice in my opinion.

Alternative choice:

or or

I think that this might be reasonable choice instead of Ghost.
- of course the best choice to run away from 5 opponents or get into the battle quickly. I just don't use it that often and I'm not used to.
- very good spell, altought I think on Udyr it might be useless - with your Blazing Stampede almost no one is able to run away from you.
- it would be very use full if you like to get to somewhere VERY quickly

Bad choice:

- I think I don't have to justify ;)
- Only Iron Mantle gives you some bonuses for having a lot of AP. And it doesn't do any damage, so your surge would give you only more "additional health". And I think your passive and Runes give you enough AS. And the other Udyr's spell that requires AP is Wingborne Storm which is not used in this build at all.
- I think it isn't a good choice, just because this spell is bad at all.

I use for AS, for armor, for Magic resistance and for AS too.

, - I think that AS is not necessary for the champ like Udyr but it is very useful for finishing your jungle very quickly and at full HP. It means that you can jungle almost all the time. - armor is of course VERY good for every jungler cause the damage taken at the beggining is very low. Useful to gank and jungle efficiently. And - of course Magic resistance will be useful for teamfights, mainly in the mid/late game.

I go 21/9/0.

First thing is that Offence Masteries give me some AD and AS, which if very effective for jungling and ganking at low level. And it gives me some life steal too - not too much but still useful. And Defense - 1st thing is that I have to have 1 point in Summoners Resolve cause it gives me additional money for using smite. And other defense masteries are good too - it is good for a jungler to have a lot of armor, and of course it is nice to have a mastery called Bladed Armor - very nice at the beggining.

I start with 1 point an Turtle Stance, then 1 in Tiger Stance, Then Turtle, at level 4 Bear and then
-> -> -> (3 pts).

I know this build is VERY strange :) But trust me - it works! I'll give you some reasons to use Iron Mantle for jungling:

a)It gives you some "additional health" when you use it
b)You have life steal without buying Vampyric Scepter
c)You can hold some lane for a long time very successfully
d)You can stay in the jungle for VERY VERY VERY long time

Even if you found that strange I tell you - try it! Start with 1 point in Iron Mantle. Then 1 point in Wilding Claw on lvl 2 is enough for your jungle at the beggining. And you will start to put points in it already on lvl 8 - not that long to wait! And of course 1 point in Blazing Stampede - useful for effective ganks. I use Wingborne Storm hardly ever playing Udyr, however I will maybe write another build with it.

The order I put in the table above is FULL - these are the only things you buy.

Of course I start with and 5 - I don't use all of them in the beggining. Then I buy (you can start with too) and then - very good for almost all junglers, and gives you 1 free ward every 3 mins. Then I buy and I stack my Trinity - first I buy for some health, then for some additional mana and then I usually buy my . Then I buy - a lot of health - very good for champ like Udyr. After Warmog - - additional damage is very useful. Then - magic resists for late game and at the end the instead of Lantern (you can buy other items for 5th and 6th place in inventory in some cases of for example: when having only AD opponents buy some armor).

And the best thing is than Udyr is a champ that can "live" without recalling for a long time. So for example there is a big chance that you will get a Warmog in the next recall after buying Trinity Force. That's it for Items.

So the core is:

I've seen this route somewhere. It is not very popular I think, but very good for Udyr.

WRAITHS->WOLVES->BLUE->WRAITHS->RED->GOLEMS->ENEMY BLUE(if they don't have a jungler)->EVERYTHING

So I start at wraiths. I kill the blue one with Smite and then kill the other ones when my Iron Mantle is on. Then I kill wolfes and you should have lvl 2 there probably, so I put 1 point at Wilding Claw. Then I use 1 potion and kill the Blue Golem very quickly with another Smite ready to use. Then i go back and wait for another wave of wraiths to respawn. I kill them and go for Red buff with another potion. So I use only 2 potions for completing my whole jungle route. Then I get lvl 4 so I put 1 point in Blazing Stampede and if necessary I go and gank some lanes - if not I jungle and collect money for my . Then you can do it on your own I think :) Don't remember to put a ward from a Lantern frequently!! And try to have blue buff all the time - you waste a lot of mana, mainly in the beggining.

These are some tips, which I think are obvious for people who play Udyr very often, but might be very useful:

Kill wraiths only with 1 usage of the Iron Mantle - not worth to waste mana.
When you're going to kill wolves - turn on your Iron Mantle once or twice - the same as it is with wraiths.
When you are going to kill the blue buff for the 1st time in the match, change your stance every 2 seconds - with your passive ( Bridge Between) you will have a lot of AS. But when you see you are out of mana - stay at Iron Mantle - you don't have any other life steal.

I think I wrote everything you need to be a good Udyr :) Hope you enjoyed it. I'll try to change every mistake I/you notice. And be aware that I'm from Poland so my English isn't perfect as you probably can see :)