Be upfront (sort of) and say you need to take a day off to take care of some personal business. If they ask what that is, you say it's personal and you will discuss it at another, more appropriate time.
Then you take your day off and think of what the "personal business" was in case the man (or woman perhaps) in charge really presses you to disclose.
You should be able to take a personal day off every so often and not have to have an excuse...that seems reasonable to me. Just make sure you take steps to have someone cover for you if need be.
Take a proactive approach. Bosses like that.
