YES
This is pretty much what I've been doing. I've lived and worked in 4 different countries so far (Poland, Finland, Norway, UK). The relevant difference is that I'm an EU citizen, which grants me a visa-free access to many job markets. Nonetheless, I have met people from other places (non-EU) who led a similar lifestyle. Here are some tips:
There are organisations which can help you get a job abroad, especially if you're a student or fresh graduate. I've used the Erasmus programme (Europe-specific) and AIESEC (worldwide) to get 2 of my jobs.
If you're willing to try a different job (at least temporarily), there are many opportunities for English teachers worldwide, and especially East Asia. You can get a significant help with visas and other formalities if you apply through an established organisation / company. A job of a teacher doesn't have to be full-time, and you could possibly work on some IT stuff in your spare time.
If you're willing to reside in a single EU country for over 5 years, you could apply for citizenship which will grant you access to the EU work market (given the political situation doesn't change).
IT experience is transferable between developed countries, so your US work experience should be deemed relevant in EU, Australia, etc., but your experience from Vietnam may be looked down upon depending on the circumstances. You should be able to get some career progression going on even when moving around.
Getting a job for an iternational corporation and moving around is possible but not very likely for a software engineer (based on my experience). Mind that even if you can transfer, it may not be to the location of your choice.
If you can get a job that's ok with working remotely, it's definitely possible to continue your usual work while living somewhere else (e.g. in a foreign country of your choice). This is, in my opinion, a better option that international corporation.
If you work a salaried job, it may be seen as a disadvantage to change employees very often. It could possibly benefit you to start a 1-person company and work as a consultant on a contract basis. In most developed countries, starting a company is pretty straightforward and inexpensive, but it will require some extra effort from you, especially if the paperwork needs to be done in a foreign language.
As a US citizen you'll be still liable for US taxes, even when living abroad. I know some US expats and they all complain about it, so be prepared.
The working language in many IT companies worldwide is English. You can get by in many countries in Europe without speaking the local language, but it can be problematic sometimes in everyday life e.g. getting paperwork done, visiting doctor, or just meeting new friends.
Visas can be problematic, though experiences vary. Plan your next step well in advance (where I'm going to move next) and have a plan B in case the target country denies you a visa (do I stay where I am and apply again? Go to another country or maybe return home?).
I would recommend that you save significant amount of money as a safety net in case something goes wrong and to get started in a new country. The costs of moving, renting your first flat in a new place, buying all necessities can be larger than expected.
Mind that your credit rating etc. will most likely not transfer to another country. You can have problems to get anything else than the most basic bank account every time you move, let alone credit cards or loans. Another reason to have a money cushion. Be prepared.
I hope you find those tips useful and good luck!