I actually drop my Aussie accent when overseas, but it takes a bit of practice. In the most extreme cases, I start developing an eastern-European tone when I do speak English. This has proven handy. ;) However, speaking other languages (French, Italian) a British tone creeps in.
Returning to Australia, it's common for locals to assume I'm British or Irish. Not deliberate.
I've deliberately put an Italian accent into my Indonesian speech - long story why, but it saved me a lot of trouble!