I've discovered a product that seems promising. It's called a Smart Pot (http://www.smartpots.com/) and it is essentially a nylon fabric pot that is filled with soil and plants. The fabric construction adds some benefits over a plastic bucket - plants tend to "air-prune", or move away from the edges of the pot as they come into contact with air moving through the porous sides. This stops the practice of girdling roots that circle the pot, and the roots become fibrous and larger, taking in more nutrients and producing healthier plants.
I decided to try these this year, and purchased two 15 gallon Smart Pots to use for potatoes. My soil is clay and since potatoes like depth, it's much easier to do in a pot. I started with a few seed potatoes from a local farmer, Marie Tedei of Eden's Organic Farm in Balch Springs, but you can always use organic potatoes that you get from the grocery store. Only use organic - the others are sprayed with chemicals to prevent the potatoes from molding and sprouting.
I let the potatoes harden a bit; in fact they were shriveled and ugly when I planted them. I cut the potatoes into sections, each with an eye or sprout, dusted them with sulfur, and arranged them in a few inches of soil in the bottom of the pot. I added 3-4" of soil on top, and watered them well.
Poof! If the foliage is any indication, I'll have some great potatoes this year. The plants were up in a few days, and are currently about 30" high. They are starting to flower, so I'm watching to see when I need to harvest, and I've been adding soil to the pot and rolling up the sides as the plants have gotten larger. I'll let you know what the final result is!