Here's one explanation in the news:
So for myself, I just increased my daily dose of iodine. But it is important not to take too much! If you take 300 miligrams (which doesn't even sound like much) you can shut down your thyroid and hurt yourself bad. The US RDA for iodine is 250 MICROGRAMS (not miligrams!!!) per day. The Japanese RDA is 800 mcg, and they have a lot less cancer than us.
I'm taking MICROGRAMs of iodine. Because of the events in Japan I added one drop of Atomidine per day, which is 600 mcg, to my supplement plan. It's a liquid in a little dropper bottle that I got for cheap at the health food store. Even if you're not sure you believe me, you might want to pick up some kind of iodine supplement before the health food stores run out. Then do your research knowing that you have the stuff already sitting on your shelf.
Make sure you check any other supplements you take to make sure that your total iodine intake stays in the microgram department. There's a little iodine in my multivitamin (150mcg), and another bit in my thyroid support supplement (225mcg), so I'm getting a total near 1mg/day. I don't know what the optimal dose is, but I feel happy with my level of intake.
If you've been iodine deficient and hypothyroid, and you start taking iodine, your thyroid could kick into gear making you warm, high energy, and making your heart go faster. So watch yourself. If you have a heart condition, be very very careful!
About iodized salt: the only research that I've run across suggests that we don't absorb iodine very well from iodized salt, so I wouldn't consider eating salt to be an adequate source.
As for antioxidants, the main ones that come in pill form are Vitamins C, E, Selenium and the herb Milk Thistle (which increases glutathione in the liver). Those are great, but your body is capable of using a great range of antioxidants, and it is smart to stock up on more than just pills. Your best source of antioxidants is FOOD!! You get a great assortment whenever you eat brightly colored fruits and vegetables. Berries with rich colors are the best, but greens and peppers and tomatoes are great too. Coffee and wine contain antioxidants, but they also have a few other ingredients with mixed health effects. Fruit juices are OK, but fresh fruit beats processed juice hands down. Frozen beats canned.
EDIT: Another variable that affects your ability to resist the mutagenic effects of radiation is your vitamin D level. If you haven't concerned yourself with vitamin D, it wouldn't be a bad time to get concerned. It is possible to OD on vitamin D, but in the winter most people can use 4000 IUs/day without going over the 40-90 optimal range. We don't know what the longterm health effects of vitamin D supplementation will be. Stand by.
EDIT 2: http://www.allisoncarracupuncture.com/?p=262 This is a naturopathic article on resisting radiation.