I owned a Mooney M20C and landed on a sand and turf runway at Dog Island, Florida, with no problems and I had a heavyweight passenger. Sadly, I no longer own that aircraft, but it had good ground clearance and relatively new rubber shock donuts all-around helped. I agree with Dean that walking a field as a precaution is a great idea, and yes, we took it slow on the rollout and taxi.
Thanks Dean for the insight. A prop strike would be my major concern as I wasn't sure how much the nose gear can compress. I have an AA1B that takes the rough surface reasonably well as long as I keep it slow and easy especially exiting off the runway to the taxiway.
The nose strut assembly of the Grummans is notorious for being intolerant of rough surfaces -- it's not hard to get a divergent oscillation going. If you're successfully navigating that surface in an AA-1B, just stick with those same techniques in the Mooney -- keep it s-l-o-w, with the yoke full aft. Only problem with the Mooney vs the Yankee is your view over the nose of the surface immediately ahead isn't nearly as good as in a Grumman.