If your sure you have all the meat and fat off of the sinew, give it a wash in a solution of bleach water. (About 1 teaspoon bleach to a quart of water)
The bleach (Chlorine) kills nearly all bacteria.
After the washing, put it where it can fully dry and it will keep till you need it. Just don't leave it where a dog might find it. My two Chessie's LOVE sinew. . . . dry or not.
You could also keep it in a frost free refridgerator, and that will dry it out, too, but it takes a long time