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This is the actual complete list of TECHNICAL Bulletins issued for Hummer H3. Some of these bulletins include other car lines, some are standard issued bulletins such as removal of rail dust.
I believe I left out the bulletin that talks about the side battery terminal, since it really does not apply to H3s.

Window Outer Belt Strip Ends are Glossy (Wash Molding) #05-08-111-004A - (Jun 16, 2005)
Window Outer Belt Sealing Strip Ends Are Glossy (Wash Molding)
2006 HUMMER H3

Engine Rough Idle During Initial Warm-Up After Cold Start and Diagnostic Enhancements for DTC P0507 (Reprogram PCM) #05-06-04-053 - (Aug 25, 2005)
Engine Rough Idle During Initial Warm-Up After Cold Start and Diagnostic Enhancements for DTC P0507 (Reprogram PCM)
2006 HUMMER H3 with 3.5L Vortec Inline 5 Cylinder Engine (VIN 6 -- RPO L52)
Built Prior to VIN Breakpoint 68125462

Trailer Park Lamps, Turn Signal, Stop Lamps or Trailer Brakes Inoperative (Tie Strap Trailer Wiring Harness to Trailer Hitch) #05-08-42-012 - (Aug 31, 2005)
Trailer Park Lamps, Turn Signal, Stop Lamps or Trailer Brakes Inoperative (Tie Strap Trailer Wiring Harness to Trailer Hitch)
2006 HUMMER H3
Built Prior to VIN Breakpoint 68104782

Service Engine Soon (SES) Light On, Diagnostic Trouble Code(s) (DTC) P0442 Small Evaporative Emissions (EVAP) Leak and/or P0455 Large EVAP Leak (Reposition Fuel Filler Housing Seal) #05-06-04-055 - (Sep 2, 2005)
Service Engine Soon (SES) Light On, Diagnostic Trouble Code(s) (DTC) P0442 Small Evaporative Emissions (EVAP) Leak and/or P0455 Large EVAP Leak (Reposition Fuel Filler Housing Seal)
2006 HUMMER H3

Service Four Wheel Drive (4WD) Light On, Diagnostic Trouble Codes (DTC) B0790, B2725, C0359 (Reprogram Transfer Case Control Module (TCCM)) #05-04-21-006 - (Sep 14, 2005)
Service Four Wheel Drive (4WD) Light On, Diagnostic Trouble Codes (DTC) B0790, B2725, C0359 (Reprogram Transfer Case Control Module (TCCM))
2006 HUMMER H3 with Borg Warner Transfer Case (RPOs NR4 or NR6)
Built Prior to VIN Breakpoint 68139888 (with RPO G94) or 68140143 (without RPO G94) - Off-Road Suspension Package

Cloudy Spot in Paint After Protective Covering is Removed (Heat Area) #05-08-51-005 - (Sep 21, 2005)
Cloudy Spot in Paint After Protective Covering is Removed (Heat Area)
2005-2006 GM Passenger Cars and Trucks
2005-2006 HUMMER H2
2006 HUMMER H3

Malfunction Indicator Lamp (MIL) On, Transmission Stuck in Second or Third Gear, Instrument Cluster Inoperative (Verify and Repair Source of Voltage Loss in Transmission Solenoid Power Supply Circuit) #01-07-30-002E - (Sep 23, 2005)Malfunction Indicator Lamp (MIL) On, Transmission Stuck in Second or Third Gear, Instrument Cluster Inoperative (Verify and Repair Source of Voltage Loss In Transmission Solenoid Power Supply Circuit )
2006 and Prior GM Passenger Cars and Light Duty Trucks
2005-2006 HUMMER H2
2006 HUMMER H3
with 4L60-E, 4L65-E, 4L80-E or 4L85-E Automatic Transmission (RPOs M30, M32, M33, MT1, MN8)


Door Ajar Lamp On, Chimes Sound or Door Locks Cycle While Driving (Adjust Swing Gate Striker) #05-08-66-006 - (Oct 11, 2005)
Door Ajar Lamp On, Chimes Sound or Door Locks Cycle While Driving (Adjust Swing Gate Striker)
2006 HUMMER H3
Condition
Some customers may comment that the "Door Ajar" lamp comes on, chimes sound, or door locks cycle while driving.


Diagnostic Trouble Code (DTC) P0757 Set, Service Engine (SES) Lamp Illuminated, Poor Performance of Transmission, Transmission Slipping (Clean Transmission Valve Body and Case Oil Passages of Debris) #01-07-30-038C - (Oct 12, 2005)
Diagnostic Trouble Code (DTC) P0757 Set, Service Engine Soon (SES) Lamp Illuminated, Poor Performance of Transmission, Transmission Slipping (Clean Transmission Valve Body and Case Oil Passages of Debris)
2006 and Prior GM Passenger Cars and Light Duty Trucks
2003-2006 HUMMER H2
2006 HUMMER H3
with 4L60-E/4L65-E Automatic Transmission (RPOs M30/M32)
This bulletin is being revised to add model years and models and change the part number. Please discard Corporate Bulletin Number 01-07-30-038B (Section 07 - Transmission/Transaxle).


Tail Lamp Housing Separates from Tail Lamp Retainers (Replace Tail Lamp Housing Retainers) #05-08-42-017 - (Nov 11, 2005)
Tail Lamp Housing Separates from Tail Lamp Retainers(Replace Tail Lamp Housing Retainers)
2006 HUMMER H3
Built Prior to VIN Breakpoint 68122692


Bumps or Rust Colored Spots in Paint Due to Rail or Iron Dust (Remove Rail Dust) #05-08-51-008 - (Nov 22, 2005)
Bumps or Rust Colored Spots in Paint Due to Rail or Iron Dust (Remove Rail Dust)
1994-2006 GM Passenger Cars and Trucks
2003-2006 HUMMER H2
2006 HUMMER H3
2005-2006 Saab 9-7X
This bulletin is being revised to change model year usage. This bulletin cancels and supersedes Corporate Bulletin Number 43-17-01B. Please discard Corporate Bulletin Number 43-17-01B (Section 08 -- Body and Accessories).

Door Latch Freezing In Cold Temperatures (Apply Super Lube) #05-08-64-002A - (Dec 19, 2005)
Door Latch Freezing in Cold Temperatures (Apply Super Lube)
2005-2006 Passenger Cars and Trucks
2003-2006 HUMMER H2
2006 HUMMER H3
This bulletin is being revised to add models and model years. Please discard Corporate Bulletin Number 05-08-64-002 (Section 08 - Body and Accessories).

Low Tire Pressure Leaking Cast Aluminum Wheels (Repair with Adhesive Sealant) #05-03-10-003A - (Dec 19, 2005)
Low Tire Pressure, Leaking Cast Aluminum Wheels (Repair with Adhesive Sealant)
2006 and Prior GM Passenger Cars and Light Duty Trucks
2003-2006 HUMMER H2
2006 HUMMER H3
with Cast Aluminum Wheels

Ticking Noise From Radio Speakers (Check for DTCs and Replace Radio, If Necessary) #06-08-44-001 - (Jan 4, 2006)
"Ticking" Noise from Radio Speakers (Check for DTCs and Replace Radio, If Necessary)
2006 HUMMER H3
with UB0, UC6 or US8 Radio
Built Up to and Including VIN Breakpoint 68195100

Sunshade (Visor) Vanity Mirror Bulb Inoperative (Replace Vanity Mirror Bulb Only) #06-08-42-002 - (Feb 9, 2006)
Sunshade (Visor) Vanity Mirror Bulb Inoperative (Replace Vanity Mirror Bulb Only)
2006 HUMMER H3

OnStar Generation 6 (Digital) System Inoperative/Has No Power, LED Light Not On (Remove/Reinstall OnStar Fuse and Replace VCIM or Follow SI Diagnostics) #05-08-46-001B - (Feb 13, 2006)
OnStar® Generation 6 (Digital) System Inoperative/Has No Power, LED Light Not On (Remove/Reinstall OnStar® Fuse and Replace VCIM or Follow SI Diagnostics)
2004-2006 HUMMER H2 prior to VIN 6H108876
2005-2006 HUMMER H2 SUT prior to VIN 6H108876
2006 HUMMER H3
with Digital Gen 6 OnStar® (RPO UE1)

Malfunction Indicator Lamp (MIL) On, DTC P0894, Harsh 1-2 Shift w/4L60-E, 4L56-E, 4L70-E Automatic Transmission (Test/Replace Torque Converter Clutch (TCC) Enable Solenoid/Harness, Auto Trans Wiring) #06-07-30-007 - (Feb 28, 2006)
Malfunction Indicator Lamp (MIL) On, Diagnostic Trouble Code (DTC) P0894, Harsh 1-2 Shift with 4L60-E, 4L65-E, 4L70-E Automatic Transmission (Test and Replace Torque Converter Clutch (TCC) Enable Solenoid / Harness, Auto Trans Wiring)
2006 HUMMER H2
2006 HUMMER H3
with 4L60-E/4L65-E/4L70-E (RPOs M30/M32/M70) Automatic Transmission