I have over the years replaced the underside of my mattress - you can get the same type of fabric at the larger fabric stores, but your best bet is to get something thicker, almost a canvas thickness (or even canvas if you can afford it) or even thick plastic sheeting. You have to go to a fabric store that has an upholstery section in order to get something in a wide enough width. It also takes at least two people (the more the merrier) to get it on right.

Karen's solution may be the best - I had never thought of that!

I have a new mattress set, too, but thankfully none of mine spend much time under the bed.

However, my new comforter that has already been through the wash about 10 times to get all the cat hair and grass barf off is another story.....