I totally understand your plight - being an Aussie I face the same accent problem everyday. But I've found a happy medium - sometimes i'll throw on a bad Yank accent to make a word clearly understood eg 'river': instead of saying 'rivah' (aus speak) i'll say 'riverrr'.
It really gets my goat when students complain to school directors that they aren't learning correct English simply because they don't get a teacher from the USA. Since I first encountered this problem I now make a point of explaining to all new students that English is an INTERNATIONAL language and that there are countless variations. I've also found that it works in my favour to explain that because there are so many variations of English, the onus is on the student to work extra hard at their studies.
I also like to point out to students that they are extremely fortunate to have teachers that aren't from the USA; they get to hear American accents on TV anyway, and hearing a different native English accent can only be a bonus. Good bit of psychology
