Crime in the US largely falls into violent and non-violent categories. Because of our culture and our media we tend to "report" more one the violent crimes, and just quietly deal with the non-violent crimes.
The largest crime your going to need to be aware of in a rural area is probably going to be theft. Same if your in the city by the way. It's far more likely that your bike is stolen then anything else, but in truth, even that is pretty rare, if you use a U-Lock.
Your two problems are going to be the vastness of the US, an the way our road system works.
The size of it
I had a family friend come from England and stay with us in Florida. They decided they wanted to see the country so they decided to drive to California. They were totally unprepared for the shear distance and what it would mean. Our country is large enough that you have different eco systems as you travel, specially east to west. Your will have Swamp, plantations, flatland, prairie lands, deserts, mountain ranges, arid zones and coastal areas (just to name a few). There are areas that get 450 cm of rain a year and others that get 60 mm.
This is also true for people. In some areas strangers are welcome and invited guests. In others they are an annoyance to be avoided. It all depends on area. Now, no one would be hostile. That's not what I mean. But in some areas you will have people invite you over to their house so they can bake you a pie, in others it will be hard to get a passer by to tell you the time of day.
Even animal concerns are going to be different. Most parts of the US, (even in cities at times) have loads of wildlife. Cougars, snakes, (brown or black) bears, alligator, panthers, crocodiles, bob cats, and so on, are very common in rural areas, some even in cities. Smaller animalas are "worse" if your going to be camping out. Raccoons, Possums, Armadillos, squirels, etc. all will have no problem getting into your food stores.
Cell coverage doesn't reach everywhere are there are large stretches of road with nothing on them. Your could easily find your self sleeping rough.
The road system is vastly different in many subtle ways. Most importantly, is the county, state, US highway, Interstate differences. Before the interstate system, if you wanted to travel across country, you would take a US highway or maybe some State Roads. But largely these roads were made to get you from specific point A to specific point B. Small towns and villages popped up along these highways. Then comes the interstate system. This system is designed to take you to maybe 1-2 big cites in a state, but largely to travel long distances between many large locations. They are not meant (primary) for travel inside the same state, though because of their size, they end up being really good at traveling between large cities in the same state.
The result is, if you're in a car, going from one rural area across the country to another rural area, you would take a county road, then a state road, then the interstate. Travel for several days, then get off the interstate near a large city, and take the state and county roads to your rural destination. For a biker, you can't get on the interstate. To do so would be illegal and life threatening. But you can travel the state and county roads quite safely. That said, you, as someone not from the area your traveling in, will have a very hard time figuring out if a road is "safe" to travel. It could be nearly empty, or it could be a main thoroughfare for that part of the state. Even more so because the same road in different areas would be used very differently.