I had an e36 through my last couple years in college and beyond. It was actually a very cheap car to run, but that's basically because it was well kept up at the time I got it and kinda rode it down into not-quite-beater status. I'll speak to the E36, since that's where my experience is and what are most plentiful in the lower price bracket (though the E46 has become very affordable as well.)
The cooling systems are basically considered a PM item on a ~5 year replacement schedule. As mentioned above, lots of plastic bits like the water pump impeller that tend to get old and brittle over time and can cause catastrophic failure.
Interiors fall apart and some bits feel quite cheap for a "luxury" car. Headliners sag, door handles can break, leather delaminates from the door cards, almost none have all pixels functional on the OBC. But the ergonomics were pretty good (IMHO), seats are supportive and comfortable, and its generally not an unpleasant place to spend time.
The super duper multi link rear suspension works great. Until every one of the roughly million bushings rots out, that is. Then they tend to get a little sloppy. Replacement gets a bit pricey when you add everything up too. Also, mounting points for the subframe are weak and need to be reinforced if you plan to do serious track work. There are kits out there for this. But when everything is relatively fresh and working as intended they really are great handling cars, predictable and easy to toss around with good feedback and tactile response.
The BMW sixes are wonderful motors, smooth as hell and quite torquey. At this point in time, the naming scheme of the model actually did match the engine displacement, i.e. 325=2.5 liters (known as the M50), 328=2.8 liters (known as the M52.) Intake manifolds from the 325s flow better and swap onto the later cars for a cheap ~20 or so peak hp gain. Cams from the S50/52 (E36 M3 motors) are a popular upgrade as well. The later motors were single VANOS (IIRC), BMW's variable valve timing system. VANOS actuators can fail at higher mileage and make noise, although I forget how bad they're supposed to be to replace. Mine had a noise that may have been the VANOS, however it was assassinated by a rogue Tacoma before I could diagnose the issue. At 207k mine did not burn or leak any appreciable amount of oil and I am confident it could have lasted well past 300k.
So, in summary, yes they can be cheap to run, if you pay up front and buy a decent one in the first place, however I would strongly recommend against buying a cheap one and attempting to rehab it. That being said, I would wholeheartedly endorse buying one. My Bimmer probably endeared itself to me more so than any other car I have owned, and this post is really making me miss it. Fantastic all-rounder that gets so many of the details just right. I will own another E36 (or maybe an E46) some day.