I jump him, he puts bombs on my head and runs away while his lane partner wails on me and my partner. I harass him, he outharasses me with bombs. I play defensive and try to farm quietly, he bombs me, my minions, and the World Trade Center. How does one beat a Zilean in lane?

Right, but if he had teammates that could just gank him, he wouldn't have an issue with him.

Basically, it's all about pushing fast and denying his early game. Because midgame with a little farming(mana regen and cooldown reduc gear) his harassing increases considerably, this is usually the time people have issues with.

Bombs!
If he's running in to drop a bomb on a player, make sure that player runs away from the minion packs. Reduce his mana effectiveness by forcing him to choose between hitting -1- player, or minions. And don't drag bombs onto other players, it's not nice.
Sidenote, I've seen Zileans with smite, they love to drop a bomb on a minion near you then smite it immediately to harass. Keep an eye out for this tactic.
Mana!
An early Wit's End or diff mana drain can shut him down. Zilean has LOTS of mana issues, watch his mana like a hawk, because his auto hits like a sock, and if he has no mana, he's literally an old guy strapped to a clock, kill him with impunity.

Spell resistance!
Right, yeah, I hate gearing against specific heroes, but a Negatron cloak will cut a LOT of damage off his bombs. But if you're gearing SR, you need to have a teammate that can compensate for the DPS loss, otherwise you probably won't put him down.

get a ezreal or ashe and he's done.

People get so scared when you throw a double bomb on a creep next to them, watch both bombs explode, and 1/3 of their hps disappear instantly.

Of course you can only do this once every 15 seconds or so, and people think "I just lost a lot of hps, it's time to run away" instead of "Hey, he just blew all his abilities, TIME TO CHARGE!" The proper counter to Zilean is to charge at him from an obscure angle (bushes) so that you can hit him with a slow. He will bomb you, but so long as you can survive a double bomb + some terror from his laning partner, you should be fine. However, learning how to deal with this setup can be an exercise in frustration.

Zilean gets really really strong around levels 8-13ish. Also he is really powerful for a brief moment at level 3, because chances are he'll hit level 3 before you due to his passive. These are times when his Q gets out of control, even with he has only a tiny amount of AP.

Too bad cleanse doesn't remove his bombs anymore lol

This is one of the HUGE misconceptions people have about zileans bombs...

The first thing everyone does when they have a bomb on their head is run away from their creep waves and hide by the turret. This is wrong, instead run on top of your creeps so the bomb kills or at least removes much of their health.

Why?

Because now you are 'pulling' instead of 'pushing'... You are forcing zilean to sit just outside his creep wave for exp near your turret, overextending himself. This is the perfect time to setup a gank with your lane partner and/or jungle.

if you're going to get harassed, bite down and trade it. the worst thing to do is get bombed, then just straight up back off every time so he gets those free hits. make him work to land bombs, and make him trade if he does land them.

if you need your lane back, you can get bombed, then sit on your creeps so he pushes, then you can call for ganks or set up a kill.