As of 2011 you can use Apache Commons StringUtils.stripAccents(input) (since 3.0):
String input = StringUtils.stripAccents("Thïs iš â funky Štring");
System.out.println(input);
// Prints "This is a funky String"
Note:
The accepted answer (Erick Robertson's) doesn't work for Ø or L. Apache Commons 3.5 doesn't work for Ø either, but it does work for L. After reading the Wikipedia article for Ø, I'm not sure it should be replaced with "O": it's a separate letter in Norwegian and Danish, alphabetized after "z". It's a good example of the limitations of the "strip accents" approach.