MUPs have their own unique set of dangers relative to roads. You have to adjust your riding to each. I ride a lot on roads -- including city streets while commuting -- as well as greenways. When riding on MUPs, I have a bell on my bike and use it frequently. If I get no reaction from the walker/runner/slower cyclist, I slow down and call out "on your left." MUPs are not the place for fast group rides, time trials and Strava record attempts -- although I will crank up my speed on trails when I can see a good ways ahead and the route is clear. I always keep my hands near the brake levers and watch children and people wearing ear buds like a hawk. The major advantage of MUPs, of course, is that you don't have to worry about cars (unless they cross roads) and that relieves a lot of stress. The greenways in my city and many places are also very scenic places to ride.
On the roads, I put a high emphasis on visibility -- on the theory that most drivers do not want to hit cyclists but often do not notice them. I use multiple head and taillights while commuting, and I wear high viz jerseys, jackets and vests most of the time. I choose my routes to avoid heavily trafficked and narrow roads with no shoulders or bike lanes. I try to be very aware and alert at all times, particularly looking for cars that might pull out or turn in front of me. I've been road cycling for 40+ years and I've never been hit by a car, but I have avoided a few potential hits through vigilance.