They are tough, tougher than H2's as you'd expect. Besides the obvious like avoid demiliterized models, the 04 is a solid vehicle without any major issues I can think of off the top of my head given its nature. There is a huge amount of information about the 04 and it's similarities/differences with other year models [06 has different engine/transmission etc.] here
04 Hummer H1 mechanical issues.