I'm coming up on 19 years on Coumadin, and I've run, ridden, kayaked and done everything else I used to do except play football for all that time. I've had two big-time falls (one over the bars with a separated shoulder) and a few smaller ones. No serious problems with any of it, except of course for spectacular bruising from things that wouldn't normally be a big deal.
This is with my doctor's blessing and encouragement, by the way--I didn't decide on my own that I knew better than those sissies.
Your condition may be different, of course, but briefly, I went into full-time atrial fib when I was 43. Chemical and electro conversion failed, and the cardiologist recommended Coumadin to prevent clots (it was still unproven and a little controversial at that time; the benefit is well-established now). Been on it constantly since. The fibrillation slows me down (I went from running a 40-minute 10K to around 52 minutes almost literally overnight), but there's nothing I can do about that. Only changes in my life have been that I quit playing in an over-40 flag football league, which I would have done by now anyway, and I quit pushing my wife to let me get a motorcycle (part of our deal when we got married was that I wouldn't ride until the kids were grown up; they're 26 and 22 now).
Check with your doctor, of course, because your situation may be different. But if you get an answer you don't like, think about a second opinion.