After a fair bit of research, I've decided to start supplementing with iodine.
Iodine is a necessary micronutrient which many people believe we don't get in sufficient amounts in our diets, due to soil depletion, absorption of heavy metals and halogens, and the fact that about 1/2 of home cooks no longer use iodized salt.
Iodine is required by many organs and glands in your body, most notably the thyroid and reproductive organs.
I'm using a product called Iosol, which is a water soluble form of iodine that also contains iodide (your body requires both iodine and iodide). One drop contains 1830 mcg of iodine, roughly 1200% of the RDA. However, many Japanese consume 13 mg of iodine on a consistent basis with no ill effects.
I took my first dose today. I'm starting with one drop in a glass of Emergen-c in the morning. I couldn't taste the iodine at all, luckily. I didn't notice much in the way of effects except for a very slight metallic taste in my mouth and slightly increased amounts of saliva, neither of which were unpleasant. Within the first hour of taking it, those effects diminished.
I'll be updating about how things are going with the supplementation. In the meantime, here are some further resources on iodine supplementation.
Two podcasts with Dr. Jorge Flechas
Jimmy Moore interviews Dr. Flechas (appx 50 mins)
Nora Gedgaudas of Primal Body, Primal Mind features a guest on thyroid and iodine
Dr. Brownstein, article on iodine supplementation
Iodine and halogens