Yeah, there is another thread that is recent on this. Check general Zx6R section or 2 wheel discussion.
I had this problem with the last Zx6 I had (it was a D). Turned out the carbs had gotten gummed up, my friend suggested an italian tune up (stark bike while parked and rev engine lots) but in the end I had the carbs rebuilt. Solved the problem perfectly.
Good luck!