Tires do ok, but it's hard to reach in to the bottom and get a couple of big ones when you want them. I've done tires, wood boxes, chicken wire but here is the two best:
If you have a little slope to your area, take a middle buster, or plow and dig a furrow running with the slope (so water will drain out one end of furrow). Lay a row of taters in the very bottom. Fill up furrow with straw. The bigger and deeper the furrow, the more room you'll have for straw so more tater growing room. Best way to do the straw is to blow it in till you have a heaping row covering the furrow.
or,
you'll just have to try this to believe it.....
find a nice spot of yard grass (stays kinda short kind), take tater eyes and lay them out kinda like a grid pattern...however big you want to water. Lay them right on the grass, about 4-6 inches apart. Cover with about 6 inches of pine straw.
When the taters are ready just pull back the pine straw they'll be a layer of taters and roots in between the yard grass and pine straw. Easy to pick up that way too....no digging.
Both ways were done with a water soluble fertilizer (generic Miracle Grow) running through the sprinklers (on the yard one) and through the drip tape on the furrow one.
Either way you want PLENTY of LOOSE straw. Rain and time will pack it for ya.
Good luck!