You can buy an Outers or Hoppes 22lr kit for about 7 bucks at the sporting goods store. It should come with a 3 piece aluminum rod, a plastic jag and a bore brush with oil and solvent with patches to clean your .22. You can even get a boresnake in .22 cal. I realize that you said centerfire, but a 22 is a 22 and the only real difference that I can see is that your bolt is much easier to remove.