University of Iowa is more of the business school in Iowa. UNI has a business school as well as Iowa State (seemed like in the late 80s/early 90s all the guys who got weeded out of the engineering program went on to major in business). I'm pretty sure that Iowa's business program is accredited, not sure about the other 2.
Generally, I always thought that that Iowa State was more the Ag/Engineering/Vet Med school and Iowa was the Law/Medical/Business/Liberal Arts school. I've worked with people who had Engineering degrees from Iowa, and they were top notch folks as well, so I'm sure their Engineering program is good, too.