From the employer side, H1B sponsorship is quite easy. They sign sponsorship paperwork and post a Department of Labor form which verifies that the wage offered to the foreign medical resident is at least 95% of the prevailing wage for the employer.
About 75% of residency/fellowship programs agree to H1B sponsorship. Most of the other 25% of the programs who refuse to participate do so due to lack of knowledge concerning the H1B visa application process or due to an obscure university/hospital policy requiring residents/fellows to obtain J1 visas.