Personally, I stay away from analgesics for most pain. Occasionally, I'll use a naproxen sodium, and less frequently acetaminophen (Tylenol). These things are not easy to avoid because they're combined in a lot of other products (I check the ingredients packaging, too). Tylenol is supposed to have no effect on coagulation (although enough of it can harm your liver).
For chronic use of the NSAIDS (naproxen sodium, ibuprofen, and good old Aspirin), you may have to have your blood tested a bit more frequently, just to be sure your INR hasn't changed dramatically. I'm not sure if the doctors are concerned about overall coagulation effects or gastric bleeding - or both. However, I won't give advice here - that's for a doctor to do. Still, if I pull my back (again), I'd probably take a NSAID until I could get something stronger or some competent medical advice.