basically:
picker.setDate(null);
Addendum:
digging into the internals of the picker and setting the text of its editor is wrong because the text is only updated in the editor but not committed to the underlying model:
picker.getEditor().setText("");
System.out.println(picker.getDate());
// output
Thu Feb 28 00:00:00 CET 2013
// output after nulling the date with the appropriate api
null