Ah, poor Zygarde...a good typing, decent all-around stats, an ability that...well, it's kind of annoying for Xerneas and Yveltal anyway. Alas, Zygarde found itself in a bad spot in gen 6, what with Garchomp being faster, hitting harder, getting Stealth Rocks, and having a more useful ability. Zygarde is still somewhat viable in higher tiers of play, but only by virtue of its support movepool. Getting Coil and Dragon Dance means that Zygarde can set up in two different ways, either of which might put your opponent in a bad spot. If you don't want to set up, Glare can cripple a foe's team, and wiht Zygarde's defenses, it should get a few of them off before fainting. Other than that, it's basically Garchomp, so just do with your remaining moveslots what Garchomp would do.
It's pretty much safe to say that Garchomp's checks and counters closely mirror Zygarde's, but on the off chance anyone doesn't know how to counter a Garchomp here's what to do.
Ice - A 4x weakness to Ice will bypass Zygarde's bulk. Without boosting, you might survive an Ice Shard, that's about it.
Unaware - Quagsire and to a greater extent Clefable laugh at setup Zygarde sets. Clefable also has that STAB Moonblast and immunity to Dragon Tail to worry about.
Status - Any variety of status is a death sentence to Zygarde, and it's too slow to avoid them a lot of the time. Zygarde has a Substitute set, but you have to get the Substitute up before getting hit.
Bulky Flying Types / Levitate Users - Skarmory can have a field day going up against a Zygarde. Landorus-Therian and Gliscor don't mind it much either. Other Flying types may have to watch out for boosted Stone Edge, though, especially with Coil's accuracy boost.
Garchomp - If you thought your Garchomp had problems with other Garchomp, try running a Zygarde against one. Zygarde underspeeds it every time without boosting, and doesn't have Rough Skin to deter physical attacks as much.
All this being said, I'm still kinda excited for Mega Zygarde in Gen VII, even if we don't get a separate game for it (RIP Pokémon Z).
The risk I took was calculated, but man, am I bad at math.