You have a case of phantom draw. You don't look for voltage though, you look for amperage. A normal vehicle will pull less than 1 amp when turned off. Anything higher than that and you will have the problems described.
What you need to do is set your DVOM to amp. The next part is tricky without a special tool but you can still do it. Put one probe on the neg cable and one probe on the neg bat post itself. Now take the negative cable off without losing contact with the cable or the bat. If you do it any other way, you can "wake up" components and it will give you a false reading. Once you've done that you can start pulling fuses. Like was said before, I would start with aftermarket accessories first as they are usually the culprit. Hope this helps.