Seems like meat always carries radiation. To avoid radiation you should eat fruit, vegetables and nuts and Sunset Sarsaparilla. basically a vegan diet.
why don't these foods carry radiation?

Well I am far from an expert but I would say its probally due to build up in the system mainly in the meats (like brahmin), plus generally plants not being mobile so if there growing its probablly not in a heavy dose areaand they won't wander into some high count area grazing heh..I think over time eating a Vegan diet in fallout would get you its just not enough rads to be noticable at first but it would build in you... Plus I think the main point is most the people are starving so its a eat and suffer later or die now kind of deal..