Please DO NOT give your dog potassium iodide, and please also DO NOT give your dog potassium iodate. Neither should be taken unless there is the imminent danger of a nuclear radiation emergency. In particular, potassium iodate in large doses affects the kidneys and the central nervous system, so proper dosing is a major issue (dosing is also a major issue with potassium iodide, mind you...). You can do MUCH more harm than good by giving either of these unnecessarily. (Note: While potassium iodide is the only one of these two approved by the US FDA, potassium iodate is approved by other countries for nuclear radiation emergencies.)
Radiation will oxidize cells, so you should be upping your antioxidants.
For humans, the best are:
goji berries
prunes
raisins
other various berries (blueberries etc-- google this; it comes up easily)
vitamin C (such as powdered C6H8O6, but regular old vitamin C tabs are fine)
For dogs:
goji berries: unknown how dogs react to this. you should avoid these to be safe.
prunes: not clear how dogs are affected by these. pits are definite no-nos, juice is definitely OK, and the fruit itself (de-pitted, obviously) is thought to be OK, but since prunes are related to raisins, some recommend you avoid these. maybe best to avoid just to be safe, but do your own research.
raisins: do not give to dogs-- causes kidney damage
Other antioxidants include vitamin C and vitamin E. Both are OK for dogs. (Be careful about vitamin E for cats; their tolerance is MUCH lower than for dogs.)
Vitamin C is water-soluble. That means it is not stored in the body, as fat-soluble ones are. Anything over and above what your body needs, you (and your dog) will simply pee out, so it is never toxic at any dosage.
That said, many people get diarrhea if they take too many vitamin C tabs at once. This is a reaction to the binding agent that holds the stuff into a nice tablet shape, and not a reaction to the vitamin C itself. This is why the powdered stuff is recommended over the tablets if you are going to take a large serving, but they are really the same thing; only the amount your body is able to take at one time is different. (You most certainly WILL eventually reach bowel tolerance with the powdered vitamin C, but you will be able to get much more into your system than if you take the tablet form.)
Google "vitamin C dogs" and you will find various recommendations about the form of vitamin C that is recommended for pets. I would recommend sodium ascorbate (C6H7NaO6) because it doesn't have that acidic taste, so they will be less likely to snub food containing it. You will find that their pee becomes more concentrated (i.e., yellower). Anyway there are various recommendations out there, so google it to read the literature for yourself.
Broccoli, spinach, and Brussels sprouts are also pretty good anti-oxidants. You should not feed too much of these for other reasons (google this if you are not sure why-- reasonable amounts are OK but large amounts are bad for dogs).
You really should not take potassium iodide unless in imminent danger of exposure, which the US is NOT in. (I totally get your point about lack of information-- believe me, those of us living in Tokyo have been climbing the walls to get more information-- but the fact is, you are NOT in a situation where you should take potassium iodide.) It is most definitely harmful to health. That said, radioactive iodine will damage your thyroid more, so if you are in imminent danger of a radiological event, that is a different story; in that case you would choose the lesser of two evils and take the pills. You are NOT in that situation in the US by any stretch of the imagination. You will absolutely do more harm than good by taking these.
I have no medical background whatsoever, which tells you something right there about the value of my comments.
Please do your own research (which I know you are doing... we are too over here in Japan!).
I do think that anti-oxidants are the way to go here, and I look forward to other people's contributions & wisdom about that point, whether pro- or anti-.
Stay indoors if a plume comes over to the US; cover your mouth with a wet cloth if you must go outside and shower upon returning (this is the advice we have been receiving in Tokyo this week). Leave clothing all in one spot when you enter the house so as not to bring the radiation inside. Limit outside activity and turn off your air conditioning units. And pray for us over here in Japan. God help Japan.