the back wheels are spinning noticeably loud
It could be wheel bearings. It could be a broken tread in the tire. Swap the tires around to see if the noise changes.
You could also have tread separating going on. What for chunking. I think tenspeed meant to say "broken belt", which would definitely cause that sound as well.
Check the condition of your tires if see no issues then while duplicating the noise drift the car left and right and see if the noise gets louder and softer if so then you have a bad wheel bearing. I see a lot of bad wheel bears in your model especially in the rear.
Looking for a Used 3 Series in your area?
CarGurus has 41,344 nationwide 3 Series listings starting at $1,250.
Search BMW 3 Series Questions
BMW 3 Series Experts