You want to scrape away as much of the meat, membrane and fat as you can, then lay the sinew out on a surface where air can reach as much of it as possible so it dries before it rots.
Back sinew (backstrap, as you call it) is, in my opinion, far superior to leg sinew. It is easier to clean, easier to dry because it is thin, easier to tear into usable sized threads, yields longer threads more generally, and every bit as strong, per unit mass, as leg sinew based on some stress tests I'd done a few years back. I personally only gather back sinew now. I do not bother with leg sinew. If all you can get is leg sinew, then I guess you're stuck with it.