I think part of the trick is to not spend all your time in one place--or at least try to find out what your options are beforehand.
When I was in Australia (September 1999), the first hotel I stayed in didn't have laundry facilities. The next hotel had coin-operated machines, so I did laundry there.
One "mistake" I made was not bringing my favorite laundry detergent from the U.S.; I use a fragrance-free/dye-free product. Such things exist down under, but they're a tad more difficult to find.
A strategy I've seen others use is simply to bring "enough" clean clothes, then mail(!) the worn ones back home marked "American Goods Returned." Use the extra room in your baggage for souvenirs.
[This message has been edited by EPS (edited 07-28-2000).]