Well, there is no country whose citizens can travel everywhere without the need of a visa. But maybe some come close to it. I guess usually those countries are from the western world, high income, possibily neutral and with only few people emigrating. Citizens from those countries possibly are no "threat" to the target country. But a bad influence can of course be its history (colonialization, diplomatic disputes, etc.), repercussions for different religions, economy, trade with "the enemy", repercussions for strong immigration laws (for example there are a few african countries that make it difficult for French passport holders to enter because it's difficult for themselves to get a french visa).
I'm thinking of countries like Finland, Sweden, Denmark, the Netherlands, Switzerland, Liechtenstein, Canada, Australia, Japan, New Zealand, Belgium, etc.
But on the other hand, what countries are you think about where you are not able to get in? If you were from a poor country, yes, chances are high your visa gets declined when they don't believe that you only want to travel. But as a westerner, that happens seldom.
For you americans I can only think of problems with getting an Iranian visa. Saudia Arabia and Somalia is probably impossible, too, for different reasons. But those countries are off limits to others as well.
I think the problem is less your passport than your will to ride a motorcycle in said country. That poses more of a problem than going there as a backpacker where you have to travel on the usual bus/train routes (at least that's probably their logic).
-Azerbaijan only lets you ride your vehicle for 3 days or you have to pay a huge deposit
-Turkmenistan requires a guide if you are there for more than 5 days (Transit visa)
-China requires you to have a guide with you - somewhere around 150$ a day
-Myanmar doesn't let foreigners participate in traffic
-Israel - have fun getting there through land borders
-North Korea - ummm, I won't have to specify that, do I?
Summing it up, the bigger problem probably is the security in some countries and being able to ride your bike.
Peter and Kay Forwood have ridden their Harley to every (or almost?) country that is. Including North Korea. I think they are Australian.