Yes... you need something! Remember, when you are dealing with any type of non-metal support system, you need something to hold the layers of foam, air or water level, so the foam, air or water does all the work. I sell mattresses and am convinced that slats with less than 3" of space between them probably is enough for warranty purposes on almost every med, BUT, even putting a layer of cardboard over the slats would help keep any foam from pushing between the slats...
If you use plywood, just seal it and treat it before putting a mattress on top of it so that no mold can build up between the pieces....