I'd think the ways they are able to invalidate Udyr should be obvious. Udyr lacks a dash and a real gap closer for one. He has to burn his flash to circumvent many problems. As to how champions like Anivia, J4, Veigar, Trundle, Karth, and Yorick can give him problems? It's pretty obvious. Anivia is one I especially hate to see when playing against Udyr, considering that she has that wall of hers on top of the rest of her kit which gives him hell. If Udyr is able to tank a turret long enough to kill anivia not only once, but twice under turret? Your team deserves the loss to Udyr for feeding him so hard.
Veigar and J4 and other champions that are able to generate terrain are ones I hate to see for similar reasons. J4 is much better than you give him credit for when played well, as he's got amazing burst and can nuke squishies while still building mostly tank. Ashe is another champion Udyr hates, due to her ability to kite him.
If you're expecting to slug it out with an Udyr, you're going to lose. Raw, in your face stats and numbers are basically the only tools he really has. Very few champions are able to go toe to toe with him. Nasus for example, is a champion able to do that once you're decently farmed and have your wither maxed out... Though in general, Nasus tends to dumpster right-click champions, so it's not specific to him. He's strongest early, and he needs to gain a lead to be relevant.
Udyr is basically the Garen of the jungle.