There's a really interesting article in today's Salon (you have to do the day pass thing to read it) about programs designed to bring fresh produce into inner cities. It's a hard battle, but there's some progress showing.
The farmers' market I go to with the most regularity is the tiny, two-stall one in Mission Hill, just down the block from my office. Although most of the people that use the market seem to be employees of the hospitals around the corner, many shoppers do seem to be local residents, some of whom pay with food stamps. Although some of the items (tomatoes, fruit) are more expensive than their grocery-store equivalents, the big bunches of greens, beets, squash and so on are very reasonably priced and provide a lot of nutrition to the dollar.
My mother works for a food pantry at her church that receives donated fresh produce from a local farm. Unfortunately, a lot of it goes to waste. It's not that people won't take fresh vegetables; it's that they won't take fresh vegetables they don't know how to cook. Many of the people who use the pantry are from South America or the Carribean. The farmer is Italian. The fashionable arugula and fennel he grows for his upscale clientele are completely foreign to the people at the pantry (and to my Irish mother for that matter, who calls me with questions: I've got this thing that has long frilly leave attached to a white stalk that kind of bulges at the bottom like a garlic, but flat - what do I do with it?). The cultural disconnects are as problematic as the logistics. But both can be overcome.