The last of its kind: a Terminus build
The potential of Terminus on Udyr is IMO unmatched, but the problem has always been realizing it in a build. This is what I suggest after many attempts:
Runes: Lethal Tempo + Triumph + Alacrity + Last Stand/Coup, Triple Tonic + Jack of All Trades.
Shards (important): Attack speed + 2x HP/level
Rush Dark Seal + Long Sword + Dagger + Boots 1. This gives us the first Jack of All Trades milestone and makes up the missing AD from greedy scaling shard runes - these are very important to enable Terminus later on.
Follow it up with Hearthbound Axe and max Q first: your adaptive damage from Jack of All Trades gets converted to AD, and you are a monster early with all the attack speed + the second elixir from Triple Tonic.
We don't need to finish Terminus now, we have enough damage and the % penetrations are meh early. Now build Merc Treads into Dead Man's and alternate Phoenix and Turtle skill points. Since we aren't relying on empowered Q/R for damage per se, make ample use of empowered E followed by R to close the gap and keep them in range while stacking Lethal Tempo (and later Terminus).
Now build Glowing Mote to max Jack of All Trades, and work on Terminus. Even before Terminus the potential for early-mid game carry is there with the combo of DPS, utility, and defenses.
You may ask why not just build Trinity instead of Terminus at this point. The answer is that Terminus has the higher carry potential. With it you can duel tanks/bruisers more reliably, and more importantly team fight, without the need to empower Q/R.
Now go for Riftmaker and max R - this makes up for all other potential MS bonuses we lack in combat (higher level E, Phage proc, etc.).
From here on, I am still testing. The three options I mostly consider for the last two item slots are Titanic (very good for the auto reset to stack Terminus + Lethal Tempo), Liandry's, and Jaksho. I have been going Liandry's after Riftmaker because Haunting Guise is by far the best component considering all the damage we have.