I'm curious about the different ways players who run Knights of the Hollow Hill overcome the slow start inherent with that agenda and get their decks rolling.  What do people to do speed their decks up after the lack of setup and how do players protect themselves during that vulnerable period?

Ive ran it a few times with a Bolton deck which should be quite viable, but I seem to do better without it.

 Opening with Valar Morghulis or First Snow of Winter is popular. If you're running a noble heavy deck opening with the power of blood and only playing out nobles may give you a slow start but it will protect you from military claim. Beyond that House Martell has several tricks, Red Vengeance is probably the best but burning on the sands could be good as well if you're not using your restricted slot for something else. At the gates is another good option, guaranteed character into play and 5 gold is nothing to sneeze at. 

First Snow is pretty much an auto-include in KotHH decks, as was S&D prior to the errata.  Manning the City Walls is really helpful too, you really only need 4-5 Armies in your deck to make it useful; free Vipers Bannermen makes up for both the gold loss and card loss (from lack of redraw after setup) that comes with KotHH.  Loyalty Money Can Buy gives you some breathing room, and Retaliation isn't bad either, especially against control decks… for ex, Lanni kneel can't really do much when you have nothing on the board and he/she has to go first. 
I don't really think Valar is a good open since you blow your only reset Plot just to kill off their setup weenies.  I agree with Shadowcat that Martell is in a good position with RV; my Martell KotHH deck has 3x RV and no other use for Influence at all… get an Orphan of Greenblood and RV in your hand and their challenges mostly fail in Round 1.  Also, anything that can be played or put into play Round 1 pre-plot followed by Rule by Decree will significantly reduce the disadvantage of KotHH… the 2 Influence bonus is free Ambush and pretty much guarantees RbD will work, 6-8 cheap Ambush cards will take care of that.  At the Gates is good too, 5 gold with a free Maester is pretty damn good.
KotHH works best out of Martell or Targ, for Targ players with a decent amount of Ambush the 2 Influence is pretty much like +2 gold all the time.  Unless you have a really weird build the 2 Influence from KotHH is mostly useless to other Houses since they don't have much use for it, although there are exceptions.
Worst case scenario is your opponent plays FoW first Round, but even if they do you're probably just going to lose 2 cards to an I challenge since you wouldnt play any characters if they have superior M position, and if you're really that worried about FoW just play Forgotten Plans as your first Plot.