I've thought about food being used as a trade item. The thing is, it is different than the other resources (which are used up completely on use). Food is a constant excess of food, so it must be consistently grown. If food is tradeable, it should only be with those civs you have good relations with, and if diplomacy drops under a certain point, that food goes back to the producer. Additionally, if the trading civ destroys a farm, it should subtract from traded food first. This wouldn't mean much, unless you wanted to build more cities, as negative food doesn't have any consequences (I feel it should), but still would make for an interesting diplomatic option.
(edit) As another thought, it could be handled much like a trade route. Perhaps with freighters, or just from the trade screen. Embargoes would stop all food trade as well as trade routes, and war would halt food trade as well. This would emphasize that food is a regular input, and not a one-time-use type of resource.