There are a few things that work very well against Zyra as a support.
Zyra is very very dependent on landing her E, Grasping Roots. The reason that Mobility counters her is that if you never get rooted, she can't properly full on engage you and she isn't nearly as useful in lane as she could be. As long as you can avoid getting nailed by her root you'll be fine.
The reason range counters her is because of how slow her E is. Since it got nerfed, it's a lot easier to dodge, and if the support playing against her has high ranges themselves they never need to get close and get rooted and all-inned. Champions like Lux or Nidalee can poke from high range without getting nailed.
CC counters her because she is squishy and very reliant on being an active part of the fight. Before she gets any health a high CC champion like Blitzcrank can effectively CC her long enough for him and the ADC to bring her down before she can do much of anything. However this can be risky because if you don't kill her early before she can get items like a sightstone to become tanky, she'll still get her full spell combo off even after being gripped and the fight can easily go in the Zyra's favor.
All in all the best way to fight her is to make sure she can't do what she needs to do, root and all-in you. So play champions that can keep her at range and never let her in close enough to mess you up.