The theory is that it is deeply aesthetically pleasing to eat what your distant ancestors ate and drank. It offers an appeal not to physical health, in the way that organic or "health" foods do, but to inner peace and a sense of history and connection.
My father is a fan of Ezekiel 4:9 bread*, which pitches its Biblically prescribed recipe of an assortment of "ancient grains", in a manner similar to a Passover Matzo, which is another example of authentic food. Razib notes an effort to make beer according to a Sumerian recipe.
* The verse reads:
Take thou also unto thee wheat, and barley, and beans, and lentiles, and millet, and fitches, and put them in one vessel, and make thee bread thereof, according to the number of the days that thou shalt lie upon thy side, three hundred and ninety days shalt thou eat thereof.