If you are flying a US registered aircraft with a US issued licence then under international ICAO agreements, countries will in general accept that as being suficient.
Some countries may have requirements for over-flight permissions. The information will be published in the GEN section of the AIP for that country.
What you need to check and comply with is the sections covering entry and transit of aircraft, as well as entry and transit of passengers and crew. You will laso need to check things like equipment required, requirements for flight plans etc.
All is available in the AIP.
If you can't be bothered to read the AIPs you could contact Jeppesen or similar organisations who will do all the work for you (for a hefty fee).
Regards,
DFC