First off, this is the first build I've made for LoL, so bear with me, and I apologize for any mistakes made in the process.

Udyr is a tanky dps that makes a great carry/pusher in 3v3 solo queue.
That's what this build is based on. I've played 75 consecutive games with Udyr on the Twisted Treeline, and once I had finalized my own personal build to the point where I was comfortable with it, I started playing more aggressively and began my winning streak.
In the last 50 games, I have won 47 of them. One loss was due to the fact that we all got disconnected, so loss forgiven. The second was someone playing Veigar for the first time combined with the 3rd played going afk right at the start. The last game I lost was because I grabbed the wrong runes and masteries by accident, which was entirely my fault.
So read on, give it a try, and enjoy!

3xI grab 3 Quints of Alacrity because Udyr has a rather sluggish attack speed at about 0.68 at level 1.
With his passive he cranks it up a notch, but for this build I want him to be faster at lower levels so I can grab the top turret in under 7 minutes.

9xGreater Glyph of Focus
This there's no reason why the AP champs should be the better casters, and with a stack of these CD Runes, I can spam my Phoenix and Tiger Stances until a Baron-buffed Sion is laying at my feet.
But really it's for spamming Tiger Stance to rip apart turrets. I don't go for kills in 3v3 unless they get in my way.

9xGreater Seal of Evasion
Since this build is based heavily on turret dives, getting the dodge percentage up high is pretty important. It's also handy in 3v3 when it comes to jungling early on. There's nothing more frustrating than when the wolves get some lucky crits on you and you fail your first jungle of the game.

9xGreater Mark of Alacrity
Did I mention that I like speedy attacks?

I go heavy on the Offense tree because this build is designed to be incredibly aggressive. I easily out-level my opposing top laning champ, since he's constantly retreating due to getting in the way of my Phoenix Stance, or getting hit with Tiger Stance when he gets too personal. I do not chase people into bushes, or down the lane, or into the jungle. I focus on the minions, and getting my minions to the turrets.

I think that the League of Legends is such a dynamic game, that having a strict shopping list of items would be at most, pretentious.
The Bloodthirster is amazing, since Udyr will be doing most of the minion killing on the lanes with his Phoenix Stance. It's just silly to pass up the opportunity. I go with Force of Nature because of the ability to sustain my time in the lanes or jungle before recalling or getting ganked.
Madred's Razors early on is a big help in jungling, as well as ripping through the lanes faster, while keeping your own minions protected from harm. I always run ahead a bit and Phoenix Stance my way into the first group of minions to minimize the amount of damage to my own minions. This allows me those extra crucial seconds at the turret to destroy it.

The items I listed at the top are slightly different from these. They're items that I have used, and will go for if I have to be a little more tank, and a little less push. Normally, that doesn't happen because I can farm fast enough to get the top tier items and break turrets before level 13.

Here's the scoop.
Go for the bottom gank as usual. If you can grab first blood, then go grab Cloth Armor and some Heal Potions. Run to the wolves and wait for spawn.
Smite+Tiger Stance=Dead Alpha Wolf. Basic attack the second one, and Tiger Stance the third. By then you can run to the lane just in time to see the enemy champ come out of the bushes and pause at the level 2 Udyr charging in with Phoenix Stance.
Zone your enemies hard. Be aggressive, but do not chase. Stay in range of your minions to protect them and to get experience from the kills.
By level 4, you can easily take down the top turret if done right. This is usually 7 minutes into the game or less.

PROS:
* Very aggressive, easily zones other champs.
* Fast as hell to level
* Great jungler, which always surprises people in 3v3
* Takes down turrets and wins games

CONS:
* High range champs like Cait and Ashe will be harder to kill
* Gets ganked a lot once the other team sees the turrets going down
* Needs a team that lets him jungle and take top lane

By level 6, you can Smite-Phoenix-Tiger-Tiger-Tiger the dragon to death in about 12 seconds.
A run through the top jungle can take less than a minute to do after level 7, and don't just hit the wraiths. Go for everything. It all counts.
Creeps come fast, but I can usually wipe the jungle clear between waves by level 9. This includes the dragon.
Keep the Phoenix Stance going when pushing lane, and kill the minions fast. You will not run out of mana very fast, believe it or not.

In Solo-queue, you will see a lot of people going nuts because they see you in the jungle. That's until you take the first turret out and the dragon, and have no deaths..or few.
Once you get the first turret out of the way, you can run bot between waves now that you can let your minions run ahead a little more without getting shot. The enemy champ will want to hang top and defend that last turret more often now, so it's going to be a 3v2 on the bot lane in your favour.
Wait in the bushes near the turret for your team to push close, and tank the turret hard. If you die, you die knowing that one more push is going to set the other team back a turret.

At that point, head back top and push through the jungle into the top inhib. Pay attention to your team on the minimap and don't be afraid to run into a teamfight. There's a really good chance you're going to be the highest level champ anyways, so it's going to end with your team in their base by the time your other two teammates are about level 11. You should be about 14 or 15 by this point.
At least that's been the pattern the way I've played Udyr on 3v3.

Thanks to everyone who helped me along at the beginning, and to everyone who had the patience to let me jungle in 3v3, even against their judgement.
Especially to those who trusted me in a leading role when we were behind 40 kills on some games, and still pulled out a win.
Thank you for reading, and hopefully we'll see more Udyr action in 3v3!