I've owned a '98 Z3M since August of last year. I've not had any significant issues other than discovering the inside rim of a rear wheel was bent, and caused a tire to fail. The Style 40 17x9 wheels aren't a cheap replacement. Luckily, the Z3 has the 17x7.5s in the front and back (the //M has a staggered set standard). There are some items I've listed below that concerned me when I bought the car, but for the most part aren't huge deal-breakers.
If they're the motorized seats, they will rock on starts/stops unless the seat is pushed all the way back. This is a fairly cheap fix if you are okay with pulling the seats out yourself and replacing some bushings, which can be had for about $25 total for both seats.
The glove box will sag. There is a guy on the z3 bimmerforums that is producing a plate to fix the sag issue. Right now he's charging $5 for the part shipped (and is probably losing money on the deal), but there's a line to get the replacement part.
When checking the top, look for a crease in the rear window. If there is creasing going on, you will likely have a replacement in your near future. The window only costs about $200, so if that's all that's wrong with the top, no biggie. It zips in, and looks pretty straightforward (I have this on my to-do list).
If the radio is locked, you may be able to unlock it, for free, if you ask around on the bimmerforums. There is a guy that will take the serial number and can get you the unlock code, if you ask nice and catch him.
The manual transmissions sometime have an issue where, on a cooled down transmission, the stick will lean to the right (between 5th and where 6th would be) when in neutral. After the transmission warms up, this will adjust back to the center. I'm told that this isn't a big deal to worry about.