I am trying to grow potatoes in tires this year so they don't take up space in our tiny garden. Last year, we grew them just fine from a bag of organic red potatoes I was enjoying... organic so they wouldn't be irradiated or anything to stop them from sprouting. They were yummy, but I think because I started with tiny potatoes, we ended up with super-tiny potatoes... :-)))
(basically, one small row gave us enough for one small potato salad! but it was yummy!)
This year, I bought actual seed potatoes (Kennebec), and felt totally stupid doing it, because it was a box of small potatoes for more money than you should spend for that amount of small potatoes.
No difference in growth (they're fabulously green and lush!), but I think if you're really farming the potatoes and maybe counting on the harvest, it is better to use the seed ones because they're screened for disease, which the supermarket ones are not.
No straw, but I am mounding them up in their tires with a combination of dried grass clippings, wood shavings and some compost/manure. No sign of flowers yet...
BTW, I have read that if you're starting with big potatoes, you can cut them up so you have two or three pieces, with a couple of "eyes" in each. But there seem to be two schools of thought and one says not to cut them because it makes them more susceptible to fungus. If you do cut them (I cut some of the larger seed potatoes), leave them for a bit to dry out before planting.