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As of today, there has only been one technical bulletin issued for the H3:
Window Outer Belt Strip Ends are Glossy (Wash Molding) #05-08-111-004A - (Jun 16, 2005)
There are a few informational bulletins, but these mostly pertain to administrative functions. (GM updates their electronic service information website daily, but I do not have the time to go into it an look every night.)
There is no mention of H3 in any of the general fix it right the first time bulletins, which just remind the technician to follow the service procedure and/or diagnostics.
In addition, not all H3s will be affected by all H3 bulletins, regardless if there is a sequence or VIN breakpoint.

As for software updates, not all software updates are broadcasted to the field via the bulletin route. Software updates are always being released via CD or Techline updates to the dealers.

If this thread is not made into a sticky, it will eventually fall back and be forgotten. So, if you think it is a good idea to have it on this page at all times, I would suggest writing a moderator.