Cincinnati: I worked as an RA in the psychiatry department before medical school, so my info is old. I suspect that the program has some trouble filling because of the location, both Cincinnati itself and the University Hospital being in a somewhat scary part of town. At the time, there were fairly many FMG's (not to say anything about the quality of residents, just that the program had some problems getting US grads). And Cincinnati may technically have fallen on the Union side in the Civil War, but it's in the deep south politically. It's cheap to live in, though, and Clifton, just to the west of UH, was one of my favorite neighborhoods I've ever lived in. The psychiatry department is fairly big and there are several research groups, the two that I know being focused on biological psychiatry and bipolar disorder. If you're interested in underserved and urban, it's a great spot. The hospital has a whole separate psych ER (but in the main hospital) and has one of the top emergency psych programs. If you're thinking about child psych, the Children's Hospital is just across the street from UH and is quite good.
Case Western: I went to med school there but didn't have a lot of contact with the psych department because I did my core clerkship at the Cleveland Clinic. Faculty weren't exactly welcoming pre-clinical students with open arms either. Like Cincinnati, the residency has some problems attracting US grads, for the same reasons (only winter is far, far, worse in Cleveland than Cincinnati). Rumor has it that the director of the child program is making efforts to attract star faculty and build a large research program. I didn't investigate because I really didn't want to stay in Ohio. There's no child inpatient unit on campus, residents go to a local home for these rotations. A new program director has joined since my class went through and I've heard that he is very intense, some love him and some are miserable.
Cleveland Clinic: the department seems to exist so that the hospital has psychiatrists to do C/L. The C/L experience was great- lots of weird cases. However, I've heard that the hospital no longer has an inpatient unit, adult or child. Assuming that the residency still exists, I guess that the residents have to go to other hospitals for inpatient rotations? Also mostly FMG's. Call was rough and I've heard that the medicine rotations are painful, but maybe it's changed since I was there.