Seaweeds are ostensibly the easiest most, natural way to get iodine. The figures seem to vary substantially though, some say that a couple of fronds of kombu should easily enough me enough iodine, but on the actual packets of the kombu I buy the numbers it gives are far, far lower. Iodine is apparantly very volatile, so I wouldn't be surprised if lots can easily be lost by any processing/packaging. I tend to soak and cook it as gently as possible. Kombu seems to be better than wakame or dulse, with nori being by far the worse for iodine.
Other seafood is also excellent as a general rule, especially haddock. Back when I was eating tinned sardines as a staple I assume iodine wasn't a problem. I don't know where specifically in fish the iodine is stored, so I'd recommend going for as close to the whole fish as possible.
I also used to get a lot of iodine from cheese, back when I was getting most of my calories from dairy (or at least I was in theory, it depends on the quality of their feed apparantly). Nowadays the only other appreciable source of iodine I get is from eggs (again, in theory, but I'm hoping my omega-3 eggs are well fed).
Iodine salt would be a good bet obviously, although here apparantly freshness matters a great deal as it rapidly loses iodine content on the shelf.
I also have a pet theory that sweetbreads (thymus gland) should contain a lot, given iodine's use for the thyroid and the fact that it's stored in the thymus in humans. I always treat my thymus glands especially gently when cooking therefore, given the volatility of iodine.