I am a citizen of the United States and hold a current passport.  What 
countries can I move to and work legally without obtaining employer 
sponsorship?

You can find extensive information on the US Department of State's
Passport Services' Foreign Entry Requirements page:
http://travel.state.gov/foreignentryreqs.html

It is less than easy to address each individual country here.  I
suppose it would be better to specify the countries you are most
interested in; that way, your Researcher can narror his research.