Hey I am really interested in trying out the bucket method for growing potatoes you found. I've always avoided growing potatoes in the past because it doesn't grow as a good companion in the garden with other vegetables, and digging them up always seemed like it would be a messy waste of the good soil I've built up.
Any alternatives that will work in place of straw, if I don't have any?
Any kind of loose, dry, organic material should work just fine. Dry leaves, dry lawn clippings, even wood chips or sawdust may work. Potatoes aren't particularly fussy about where they grow. It may even be possible to use shredded paper, but I avoid this because many inks contain toxic heavy metals that can be absorbed by your plants and passed on to you.
You could also use dirt, compost, or even potting soil, but I've found from similar methods that these can get packed in quite tight and be hard to dump out.