MSF or VSO are options.
A friend of mine just got a job as a doctor on a cruise ship, if you fancy that- you work 4 months on, 2 months off, alternate days on call, tax free salary (about £50k a year). I’m not saying that the work is that exciting (basic primary care with the odd emergency), but the 4 months holiday a year give ample travel opportunity, as well as the chance to get off the boat on your days off.
Most developing world jobs require you to have a portfolio of skills- obstetrics, surgery, primary care, emergency medicine and anaesthetics. You’d be working single handed so you need to be able to WORK single handed- a typical American residency might not give you a broad enough skill set.
The BMJ (and, I assume, American journals) often have advertisements for foreign jobs. Tristan de Cunha is currently looking for doctors, for example.
The UK usually requires overseas Drs to pass the PLAB test- should be no problem for an American graduate who has passed their boards. An exemption would be granted if you can get an appropriate sponsor.
See here.
Proving your competence won’t be an issue- getting a visa might be. You’d either be on a Tier 1 or Tier 2 visa, and there are restrictions on which jobs you can have. See here. You need to have proof of financial solvency and high level skills to get the most points, and you won’t be given a full training post, but rather a service provision or short term position.
If you’re looking to come to the UK to study the Liverpool School of Tropical Medicine does postgraduate courses that would be useful if you do decide to work in the developing world, and a student visa might be easier to get (although you wouldn’t be able to work as a Dr while you studied).
Most EU countries have similar requirements regarding language and professional competencies and visas.
If you want specific advice about the UK, I can give you that.