As far as I know only bikes registered '07 onwards are exempt. Prior to that you pay the same as a car.
It is based on declared emissions, not strictly the age or euro rating. Age/Euro rating could be considered a guideline for vehicles that will be reasonably guaranteed to be exempt as that class meets the emissions requirement.
If you get your bike or other vehicle tested/certified and can show TfL that it meets their emissions requirements, then it can be exempted. DVLA will need to add the emissions figure to your registration, and then TfL will exempt it from the charge.
I think I read recently that if you are willing to do so, you can even apply to TfL to exempt an entire model of vehicle rather than an individual, if you can show that model meets the requirement.