Here's how you counter the "OP" champions in URF
With a few exceptions, pretty much everyone is either playing poke champions or assassins (or both, looking at you
).
Since you pretty much know that your team is going to pick a few poke/assassins, you really don't need to do the same.
So, what counters poke? Hard engage. What counters assassins? Tanks.
That's right folks, bring the tank meta straight into URF mode. Pick your favorite CC heavy, hard engage tank and go nuts. And before you say that playing tanks in URF isn't fun, imagine the cathartic release when that Zed can no longer play the game because there's an Alistar sitting on his face, perma-stunning him in a corner of the map. Seriously, what's he going to do? He can't kill you. He tries to poke, you just engage and watch him run away (or try to pull some LCS big play, at which point you just sit on his face until he suffocates).
Bear in mind, depending on your pick, you might have to play it safe, even hug your turret, for the first couple levels, but it doesn't matter since you're not going to lose any CS/Exp since everyone is just shoving the wave like a madman. Not to mention you only have to be in lane for like 1 minute at a time, thanks to the TP cooldown. Buy a refillable pot to save some gold. Roam around and kill some people.
Here are a couple ideas to get you started:

Also, some proof: Sejuani, who's on free rotation and Shen
Happy Hunting
EDIT: More suggestions from the comments
Her W destroys mobile champs, and you can just ult people around for fun. When I mentioned pinning people to walls, there is no one better.
If you want your opponent to have a terrible day, look no further.
.
2- Obtain level 6 and
.
3- Bait him into ulting you.
4- Once he does, ult and burst him down with your abilities and eventualy you'll kill him.
5- If he buys and uses QSS, just zhonias and hope you can still outplay him after the active's gone.
see this champ only counter is wards and she can still destroy them.