The first strop I ever tried, was a 4-sided strop 'bat' (3 grades of leather & compound, plus a stone). Had a handle at one end, and came in a paper-lined slipcase. The compounds used were different colors (black, red, white), so the paper lining had been colored somewhat, for each side. I was always careful to put the block back in the slipcase so the colors matched up. When using it, I quickly developed a habit of using it somewhat upright, at an angle, held by the handle at the upper end. I never (or seldom) laid it flat on the counter, as I was always concerned about dirt & such.
I don't use that one anymore. Prefer to use single-sided strop blocks, which I've made for myself. Even so, I've become less concerned about the 'PERFECT' polish, so I don't worry as much about extraneous dust. I usually just give it a quick wipe with a paper towel, or even swipe it across my jeans before using it, just to get rid of the BIG particles (I can usually feel those, like tiny speed bumps under the blade). If I were really concerned about the dust or other cross-contamination, I'd likely keep it in a zip-loc, as suggested earlier. And if using a double-sided strop, put a clean (new) paper towel underneath it, if laying it down. Even at that rate, it won't take long for the dust to settle on 'em. It'll getcha' eventually.
I've also been experimenting with using alternate media for stropping. I've found, with compounded stropping anyway, a lot can be done with a simple sheet of clean paper (printer paper, for instance) with compound applied to it, laid atop my regular strop block. If it gets a bit too dirty for my tastes, I just toss it. Another possibility is stropping on balsa, or other wood. When it gets dirty or loaded up, just sand it clean & start over (which works pretty well with compounded leather, too).
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