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I'm a Brit who came over here 12 years ago on a work visa and I can tell you that you want to concentrate on the big corporations with large legal departments and preferably a couple of overseas offices as they'll be most familiar and comfortable with the process. 3M, Thomson West and Cargill are probalby the best places to start. You could also try Target, Wells Fargo, Daytons/Marshall Fields, Ecolab, General Mills, Medtronic, Northwest Airlines, Best Buy, United Health Group and Xcel Energy.
BUT the thing with visas is that the company doing the sponsoring has to be able to prove that you are the only person who can do the job and that can be tricky as some companies find that very daunting. However, I'm quite sure immigration lawyers with years of practice know all the right terminology to fulfill that requirement so don't let it put you off -- that's why companies with this experience are good to approach.
Why not also try some UK-based companies and see if they have offices over here? You also want to take a good look at your experience and education levels. If you can bring skills that are rare over here, your chances are better. Try and do some research on where the skill shortages lie, eg; medical, teaching.
One nice thing about Minnesota is that with its still relatively small immigrant communities, your chances are probably better than if you aimed for somewhere on the East or West coasts. Let me know if you have any other questions!
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