We've grown potatoes over the last few years in traditional rows, in tires, and 'Tater Towers'((broken link removed)). The only thing we haven't tried yet is making wire fence towers (like a compost pile).
We've tried all the fancy potatoes, reds, blues, various whites, but now mostly just grow Red Pontiacs and Kennebecs. Kennebecs (and other Keeper varieties) can keep all winter in the basement.
All methods work to grow potatoes.
Traditional rows:
Easy to plant, easy to hill. Hard to harvest...lots of digging stooped over in cold dirt(we don't harvest till around thanksgiving right before the ground freezes solid). Supposedly, that row in the garden can't be used for taters (or tomatoes) for a couple years to avoid disease. Yields ok. Don't need to water as much as other methods. Takes alot of nutrients out of the soil, so taters should be the last crop in a rotation before going fallow for a season. I don't use method this anymore as we don't have enough room to let a row go empty for that long. In the other methods described below the tater dirt goes into the compost pile for a couple years to recharge nutrients.
Tires:
Used tires are free. I was a little concerned with gunk leaching from the tires, but read enough about tires used in gardening(google it) to feel comfortable that I'm not getting anything but maybe zinc which is beneficial in potatoes. Just scrub them real well, rinse, scrub, rinse. I washed ours with Simple Green.
In tires it is easy to plant, easy to harvest, can be done anywhere(on the driveway, poor/rocky soil, etc). When done, harvesting is easy as picking up the tire, dropping on a tarp to empty the dirt, picking out taters. Yields are okay. Don't believe what you read on the web and bother stacking more than a couple tires tall... we've only ever found potatoes in the bottom or 2nd tires. None ever in the third. Needs to be watered more often than in rows as the towers dry out quick.
Wood Towers:
This was our first year with towers. Cost $ to buy wood if you don't have it, takes time to build. Yield was poor, harvest is easy. I won't do this again.
I tried one other method this year. I filled in the sides of two large stones with half rotten old landscaping timbers and filled with dirt. The vines grew okay, I haven't harvested yet to determine yield.
see pic:
http://1.bp.blogspot.com/_gXZ1lIGFdLY/Su94KIpvBXI/AAAAAAAABB0/kHK1f2pZoGg/s1600-h/0616090658.jpg
Summary:
I think we are going to switch to 100% tire-taters from now on because I can plant in otherwise unsuitable places(on top of rocks, in the driveway, etc), and harvest is super easy. Many of the local farmers have also switched over to tire-taters for the same reasons unless they have machinery to harvest traditional rows.
Here is an article from the Oil Drum about growing potatoes.
http://www.energybulletin.net/node/50555
Economics aside, I get a huge amount of satisfaction knowing that we now grow almost 100% of our potatoes for a year. It's the only staple crop that anyone can reasonably grow almost anywhere.
As an aside, my favorite meal of the year is venison tenderloins pan fried in butter, with just-dug red potatoes, green beans or braised chard from the garden, and homebrew. There is nothing better.
