i do agree, getting time off for interviews can be a tricky situation, generally most companies require at least one weeks notice for holiday so that would be short notice.
I don't know the full situation so don't know if it would wok or not but how about trying to get the interview booked so it would fall into your lunch break, or speak to the people at the new place and ask them if the interview can be arranged at either end of the day and try and get away early or say your going to be late in on that day and you'll make the time up.
Your new employer should respect and understand that you have to fit the interview round your current job and should see this as a good thing, I'd think twice about taking somebody on who was willing to take a "sickie" from their current job to attend an interview as it doesn't show very good commitment or loyalty.