Check my posts #516 and #537 to see if that gives you any better ideas. If you're going to build 1" frames, I think you could try using 1x3 furring strips like I did. I'm not sure if you cut them down to 1" and stood them on their edges if that would be sturdy enough. If you lay them down flat, you'll need to add an additional layer of wood to make them 1" thick but that should be easy to attach.
Or two layers of 1/2" plywood would probably by fine too. I don't really see the need for MDF, even to get the beveled edge which I did without MDF.