I thought I might play devil's advocate just for the sake of a different opinion....
I went to UT so I am not that familiar with UH. One of my best friends in my med school class though went there and absolutely hated it. She was in the eight year program that guaranteed her acceptance into Baylor COM. I don't know all the facts but she eventually was not admitted to Baylor. She claims the program director did her some major injustice. She had a near 4.0 GPA and broke 30 on the MCAT but still ended up at our current school, which for all intents and purposes is a good school, but quite frankly is less than what she is capable of. She claims that it was her decision to go to UH, and their inability to command the respect of medical school addcomms that kept her from going to schools like Southwestern, Emory, etc.
Like I said, I know very little about UH, and none of these opinions are mine. However, I will reitierate that I do believe that the undergraduate institution attended does play a major role in what medical school you eventually get accepted to. If all you want is to get into A medical school though, and are not that picky about which one, then any undergraduate school will probably suffice so long as your MCAT score legitamizes your GPA.