I don't know how much this is really going to add to what is above, but I've worked on 2 other continents now (one job being in Iraq on a US military base, so that only sort of counts), and interviewed for jobs in both Amsterdam and Singapore.
It certainly IS possible to have someone else pay you to travel.
Both Amsterdam and Singapore have business cultures and laws that encourage the hiring of non-citizens.
New Zealand and Australia also seem to be fairly easy places to get a work permit if you're in Software Dev or IT.
They are also beautiful places to visit. If I were going back down under I'd aim for NZ over AU, but only by a thin hair. Both are really cool places.
One thing you might want to try is to get into the Defense Industry--if you've got a clean background and don't use illegal drugs it's fairly easy to get a secret or top secret clearance. There are then a reasonable number of jobs in Europe (Germany, a little in Italy), some in Australia etc. Check Clearancejobs.com
I didn't really care for working in that sector, but I'm a Unix Admin, not an "Engineer", and some people love it. The nice thing about it is you usually get paid US style wages plus cost of living, they USG moves your stuff etc.
Also if you know French and can get a secret clearance you can look at NATO.
Finally, if your parents or immediate grandparents were European Citizens who immigrated you might be able to get citizenship in their original country and hence EU citizenship, which would mean you wouldn't need a work visa. If you don't need the work visa, then f* it, learn to bar tend and go travel. Good bartenders are in demand almost everwhere.
Oh, and someone above mentioned a "working holiday" visa. Those are great, but at least in AU weren't the sort of thing Software Engineers would come in on as the expectation is you're going to be working part time and holidaying part time, getting a part time job to extend your finances as you party, and move around a bit (and there is a big culture of that in AU, you work a waiter job for 2 months in Brisbane, then piss off to Melbourne for a while, then maybe to Alice for the winter and etc.)
A software engineer isn't usually for a month or two (at least) after getting all his tooling set up (from desktop to tool chain access etc.) and that can take anywhere from a couple days to a month in some really bad shops. As such you might be able to get work that way, especially short term or small project work.
 I had one place I worked for 6 months and the mangler wouldn't approve me getting added to the production authentication domain. Had to ssh as me to the bastion host and ssh to production servers as root. No, really. Quit laughing, it wasn't...no, it is funny.