*looks around, realizes I should probably answer this*
There's one thing that makes concrete structures feasible: rebar.
The addition of steel with concrete actually makes concrete surprisingly ductile. It's really bizarre... The stuff can flex a lot more without breaking than you'd think. Go YouTube "Tacoma Narrows" and watch the road deck go into cyclic torsion. (Yeah, it eventually collapses, but that's a concrete road deck that's flexing until it *does* go. It's pretty sweet.)
There are a few different failure states that concrete structures go through before ultimate failure, and they're easier to visualize if you imagine concrete structures as being made of something like pretzels (concrete) with an inner core of rubber band (basically the rebar). The first state is pre-cracking action, where it's just the concrete acting. In our pretzel-rubberband model, the pretzel material itself takes all of the force and the rubber band is just chilling, not doing anything. The second state is post-cracking action, where the concrete has cracked but is sticking together and the steel rebar cage is brought into play. (The pretzel has cracked but is providing some rigidity for the rubber bands, which are stretching and giving in response to the forces.) This is a really ductile, flexible state, and it really doesn't take a lot of cracking (just really small cracking, some maybe 1/16th or 1/8th inch cracks and a lot of hairline fissures... which *all* concrete buildings have, to a certain extent) to get to this state. After it's reached this state, it's still more or less okay, and doesn't require major repair.
The third state is where the steel starts to yield and undergo plastic deformation, and that's where these concrete structures start to fail. It takes a lot of energy to get to this point, and concrete structures in seismic regions are designed to go through a *lot* before they get to this state. (In our model, the rubber bands are stretched to capacity, they snap, and the whole thing comes down, but like steel, it'd take a heck of a lot of force to snap those rubber bands in relation to the pretzel that surrounds it.)
So basically, we as structural engineers *can* do it. We've figured out what needs to happen in order to make structural concrete work in seismic areas.
Couple other points... You might be like, well, yeah, it *can* work, but you end up with a ton of cracks after a really big earthquake. I'd in turn answer that philosophically, engineers don't really care about cracks after earthquakes. We try to minimize damage during the smaller earthquakes, sure, but during large earthquakes, we don't give a hoot about cracking. Our goal is *life safety*. So long as the thing stands up long enough for all the people to get out safely, we don't mind scrapping the building. (In order for it to get to that "we don't mind scrapping the building" stage, it'd have to be an 8.5 or 9.0 earthquake... we're talking really big ones, here.)
Why not just use steel instead? Several reasons...
First, concrete's cheap. It's basically a little tiny bit of steel being held together by superprimative rock-glue. The concrete doesn't do a lot structurally except provide some bulk to build out of, and we primarily rely on the steel to do all the cool force-dissipating jazz. Steel alone is far more expensive.
Secondly, the "lead time" on steel is a lot longer. If I have a newly-approved design for a concrete structure, I can buy some wood and get my workers together and start nailing together formwork tomorrow, and I can be pouring by the middle of next week. With steel, I have to send the plans to the fabricator, who has to order the steel to spec from the steel mill, and do all the detailing, and send it back to the engineer to get the details approved, and blah blah blah... Takes months to get that project started. (Once it starts to go, it goes faster than concrete, but for a mid-rise building, concrete tends to win the ultimate race.)
Finally, steel doesn't work all *that* much better than concrete does in seismic situations. There was a *huge* controversy in the welding industry right after the Northridge Earthquake in... I think it was 1994. It was, before Katrina, the most costly natural disaster in recent American history... Hundreds upon hundreds of structural corner welds of beams to columns just flat-out popped. Ruined the columns that the welds were attached to, in a lot of cases. Didn't cause any fatalities or collapses, but the damage to the structures was just tremendous... And it had to do with the fact that the welders used things called "backing bars" to give them a substrate to weld stuff too... Long story short (too late), it was one little tiny detail that was previously held to be a perfectly acceptable thing that ended up, because of the dynamics of that particular earthquake, being incredibly damaging to a wide range of structures. (If you'd like to see some pictures, google image search "northridge weld fracture" and look at some of the things that pop up...)
Anyhow, that was probably more answer than you wanted.

Let me know if something needs more explanation!