Do psychiatrists have educational requirements other than an MD? Most pre-medical students take psychology, biology, chemistry or something science-related, in addition to the classes needed as prerequisites for medical school, for their undergraduate degree or even for a master's, but is there any particular degree besides an MD that someone who wants to be a psychiatrist needs to take?

For example, is a business, philosophy, english or political science major who has taken the required classes of medical school who later finishes medical school eligible to train to become a psychiatrist? I'm curious as to how such a person would get the psychology-related training and knowledge. Is it in the training after medical school? Is it related to medical school electives?
The only required degree to become a psychiatrist is a doctoral degree (M.D.). Using your example, yes those people could train to become psychiatrists.

After passing the Medical College Admission Test (MCAT) and going through med school, most students undergo a psychiatric residency (aka. clerkship) where they are trained in different medical centers (hospitals, health clinics, psychiatric wards, etc.). This allows the prospective psychiatrist to gain "real world" experience. After a psychiatric residency, psychiatrists must then take an exam issued by the American Board of Psychiatry and Neurology. If they pass this exam they are certified and can then legally practice as a psychiatrist.

Source:  http://study.com/articles/Psychiatrist_Educational_Requirements_for_Psychiatrists.html

Edit: Med school teaches you theoretical psychiatric knowledge whereas the clerkship will teach you how to apply it. The clerkship could also train you in specialty fields.