You could use untreated plywood with holes drilled in it for drainage, but I think landscape fabric would work...and probably be cheaper. Cardboard is a good alternative, although it breaks down over time.

Use whatever you have on hand and can afford. :)

I would agree that weed cloth attached to the bottom of the bed is the best alternative. For heaven sakes don't put hay down there. I've ranted about it in another thread recently, it's great for mulching to stop erosion where you have just dug and/or planted grass, but it is loaded with all kinds of seeds (generally grasses and strong, aggressive, weeds) and bugs. Also, if you have gotten hay that is being sold as mulch it probably has other problems that keep it from being sold for more money as feed. These could range from prickly weeds, and their seeds, to mold and mildew if the hay was baled damp (wet hay bales can also spontaneously combust, but that is a different class of problem...).

HAY is very different from STRAW!!! Hay has all the seeds, straw does not (you may find a few in there here and there, but it has like 5% of what hay does!) I mulch walkways with straw, and use it in my compost piles, and mulch with it and have NO problems. The one time I used hay by accident (thanks DH for picking up the wrong thing!), and had spread a bit of it around before realizing how many seeds were really in it, we got a TON of hay weeds there. But they are easy to pull, and we make sure we only get STRAW!!! Good luck!