Since I have owned it, nothing has gone wrong. I've done basic maintenance, new tires, brake pads and a valve adjustment, but otherwise she starts every time and goes without any fuss!
The speedometer broke during my test ride when I was considering buying the bike, but a replacement gauge cluster from ebay ($80) did the trick, and it's been fine ever since.
The headset bearings are beginning to show signs of wear, but at 30,000+ miles on such a heavy bike, this isn't really a surprise.
I have enjoyed every mile on my ZZR. It's a long and heavy bike, but Kawasaki did a great job at making it feel nimble and not nearly as big as it is. Around town in traffic, it can be a little bit of a chore, but once you hit the open road, the bike really comes into its element. Hours of high speed riding in the hills and backroads are effortless on this bike.
The engine responds instantly at virtually any RPM, and thanks to the fat torque curve, most situations don't require a downshift to get going. For being a sport touring bike, this thing accelerates surprisingly fast. As one reviewer has already said, will gladly do 180mph+ and click off a 1/4 mile run in the mid 10's. Even at 100mph a twist of the throttle produces instant and vicious acceleration... Hold on tight! The real beauty, though, is that you don't have to spin the engine very fast to get moving at an illegal pace. It's perfectly happy loping along between 2500-4500rpm, which is what Kawasaki was aiming for. It's a big engine, and it makes a bunch of power up top, but despite having fairly large carburetors and wild enough cams to breathe efficiently at 10,000+rpm, it's still very civil and torquey at low engine speeds.
Suspension is firm, but still fairly comfortable on most surfaces. At high speed on the highway it really shines, and it's very planted through big, sweeping corners. The brakes are powerful and easy to modulate, but the stock pad compound didn't have enough bite for my liking, so I switched to the EBC HH pads and I have no complaints. The stock riding position is a little sporty for longer rides, but with 1" Gen-Mar risers and a Corbin seat, it's pretty comfy.
The ZZR tends to "suffer" from little bit of buzzing in the handle bars right around 4500-5000rpm, which is right about where the engine speed winds up at 75-80mph. This seems to be the most common complaint about these bikes, but it can be mostly remedied by using different grips. Other than the little bit of buzziness, it's a great bike for long distances, and the bodywork does a very nice job of keeping you from getting beat up too much at high speed.
I've ridden mine as much as 650 miles in a single day through the mountains of Colorado, and at the end of the day I wasn't crippled! If I had been on a literbike, I'm sure it would have been a different story. I was also able to average 53mpg during that trip of mostly 70-90mph highway, and a couple of blasts well beyond that. Around town I average about 40mpg. I live in Colorado, so sometimes I can't ride, but as long as the ground isn't covered with snow, I try to ride my bike every day, and it never gives me any trouble. The ZZR1200 is somewhat of a well kept secret.