It's all in the question I think. Assume varying flight times, but one long distance flight of more than 12 hours.
As requested, an addendum to Torben's answer as regards to cosmic radiation:
The exposure due to cosmic radiation is actually about two chest X-ray images per flight, or 40 uSv, depending on flight height, latitude, and length. At 10000 flights, there would be definitely apparent effects, known as 'deterministic effects' because they can be traced directly to radiation. So unless the OP is flying in the 100s of hours per year, they should be OK-- but he did ask for how many flights would be unhealthy, not if it's a concern :)
The problem with radiation exposure comes with discerning the cause of the non-deterministic effects. Deterministic effects are far more straightforward: if you receive a particular dose of radiation in an single exposure, or if you receive enough radiation over time, you will develop problems. For instance, if you receive a certain dose of radiation to your eye, you will develop cataracts (as discussed here). If you read that carefully, they state that a single exposure of 5 Gy (Grays, same thing as an Sv, or a Seivert) will cause cataracts, or a 'fractionated' (or received over time) of 8 Gy, or a relatively constant background dose of > 0.15 Gy/year. A single plane flight is 40 uGy, or 0.00004 Gy. You would basically have to live at 30k feet to worry about this particular radiation effect.
Non-deterministic effects are hard because tracing the exact cause of the problem is tricky. Say you develop lung cancer-- could it have been because you took 80 flights last year, or because you're genetically predisposed to it, or both? What is the relative risk of 80 flights as opposed to a year's exposure to second hand smoke (or first hand, for that matter)? That's a harder question to answer in an individual, because you can't do controlled experiments. What you can do is look at large exposures of radiation (Hiroshima, Nagasaki, Chernobyl, and now Fukushima) and trace their effects over time. For instance, within 5-8 years, people near Hiroshima had more cases of leukemia, by percentage, than those who lived elsewhere.
If you look at the population of Hiroshima over the course of 5-55 years following the bomb's detonation, you'll get a view of how much dose leads to leukemia, as is done in this story. You have to get into the significant fractions of a Gray (0.1 Gy) before you see more than single-digit risk associated with exposure. Note that that's at a single exposure; plane flights are fractionated, meaning that you'd get less exposure at a single instance, but more over time.
Upshot: If you take enough plane trips to get ~0.1-0.5 Gy of dose in a year (say), you could see some problems arising from radiation exposure. ~At 0.00004 Gy per flight, you would need at least 1000 flights before you should be worried about cosmic radiation exposure.
I don't believe that there are any particular health risks specifically associated with flying.
What factors do you consider as health-related?
Noise level? Airplanes are loud, and a long flight will be a nuisance to anyone's ears, but the noise level is nowhere near a medical concern.
Air pressure? Healthy ears can equalize the air pressure in the cabin. The changes in air pressure are so small that they can't affect health directly (as opposed to, say, divers' decompression sickness).
Air conditioning? Recycling the breathing air of hundreds of passengers can't be healthy, some say. Keep in mind that the air is thoroughly filtered, and that outside air is added. You're not sitting in a closed system for 12 hours.
Cosmic radiation should not be a concern, according to this question:
Airport X-ray security? The airport scanners' radiation danger "is believed to be negligible for an individual. If 1 million people were exposed to 520 scans in one year, one study estimated that roughly four additional cancers would occur due to the scanner, in contrast to the 600 additional cancers that would occur from the higher levels of radiation during flight." (Wikipedia)