I did this past year. It actually turned out pretty well. Basically I started with one of those 30 gallon bags, put in about two cubic feet of soil, leaves, compost, etc. I planted about 4 or 5 eyes. To keep the bag from 'flattening out' over time, I made a cylinder of chicken wire and wrapped it.
As the plant grew, I unrolled the bag more and added dirt. Towards the end of the summer when the plants seemed to "less vigorous", I opened the first of 4 bags. I got more potatoes than I expected. Then every couple of weeks I opened up another bag for more taters.
I'm sure there is a better way of doing it than what I did. One thing I thought I did right was adding quite a bit of dried up leaves. I didn't want the soil to pack down because I wasn't sure if packed dirt would keep the taters smaller. The leaves seemed to keep the dirt light and 'fluffy'.

Anyway, garbage bags and taters are cheap... what have you got to lose by giving it a shot?

Skycopp:

Did you put the wire inside the bag?

Brits - and the Irish - are big on bag growing - do a search on Youtube and you'll find them growing potatoes in bags. Here's a link to one example - he has quite a few videos on potato bags.

Here is a link that might be useful: potato bag how-to.

Charlie,
I put the wire outside the bag. The only purpose of it is the help the bag stay in a standing form and not flatten out.

Good luck and post pics.

How is the quality of the potatoes when you grow them this way? I wanted to try it but was afraid of ending up with a lot of eh taters instead of a few good ones. Do they turn out as well flavored as those grown in the "traditional" way?

I did some research and apparently as long has you have them in good soil, plenty of sun and water, there is no difference. I grew some tomato/potato grafts in the traditional way last year that were interesting, but I was not impressed.