Bears: Do not store or prepare food, or eat, in your tent. Store it properly (bear bag or bear lockers). Stick to designated campgrounds in National Parks (which is required by regulation). If you follow posted signs in bear areas, your chances of being eaten by a bear drop from unlikely to really, really remote.
Bison, elk, moose: read the brochures you get when you enter Grand Tetons/Yellowstone, and follow directions.
Lions: There have been a few confirmed sightings of mountain lions in the east in the last few years. If you see one, you're really lucky.
Dogs: I'm a fan of Halt! pepper spray on a handlebar mount to deal with eastern Kentucky dogs. Practice a time or two so if you're surprised by a charging dog, you can deter it by squirting in the right direction. Get off the bike for a better aim if you have time. Mount the Halt! in central Virginia and keep it available through Missouri, just in case.
Turkey: make sure your heart is in good shape, because if one moves in the ditch while you're passing, it'll scare the **** out of you.
Snakes: stay out of brush, and if you hear rattling, go the other way.
Turtles: if you see a green turtle with a long neck and tail, stay away from that snapping turtle.
Cows: if you can get past them, see if you can shoo them back into the fence they came out of. Make sure you close and latch all farm gates, not that you'll have much call to go through them in the first place.
Recreational Vehicles rented by people who drive small cars back home: these are the worse killers in Yellowstone, and don't make the news because they're so common. Wear bright colors, try to get off the road by late afternoon, find something else to do if it's foggy.
Despite this long list of things to worry about, don't worry about them. Deaths are so rare they make the news in Britain, apparently. Injuries are probably more common, but there's a large number of people who ride across the country, go visit the highlights of the tour, etc., that come out of it just fine. If I had to make a list of things to worry about, it would start with how to drink enough water, how to ingest enough salt, what to eat, where to sleep, have I pumped up my tires lately, is there enough brake pad left, etc., etc. Lions, bobcats, and bears are right above how do I secure and transport that half pound gold nugget I might find lying beside the road on my personal list. Hey, it might happen!