There are lots of ways this is done - some people have to take unpaid leave, others use educational leave, others are allowed to use their accrued sick leave, others still are forced to use vacation. In some surgical subspecialties I have heard it is common for residents not to get anytime off and they have to use the professional meeting to interview with all the fellowship directors and never actually go and visit the hospital before hand; in others the residents get a lot of time off (like a month) just to interview for competitive fellowships.
My opinion is that programs should want their residents to get jobs or match into the fellowship program of their choice, otherwise what the hell is the point of doing a residency. To that end they should provide time off in order to interview for as many interviews as necessary (though of course there should be some limit). As a resident I was allowed unlimited time to interview for fellowships (and did not have to use my vacation, and was in addition to my 15 days of educational leave, and as a fellow I was also allowed time off (I took about 5 days and was granted additional time off for interviews I cancelled) for job interviews and did not have to use vacation (and this was in addition to 10 days of educational leave).
As a psychiatry resident, given it is an outpatient heavy specialty and you don't need anyone to cover for you except for refills etc, you should easily be able to cancel clinics and there is no reason they shouldn't allow you to take time off and should not require you to use vacation. It is a black mark on a program if they dont make reasonable accommodations for you to interview.