All cases are quiet - it's the stuff you put inside it that makes the noise.
Steel cases are best, but a high quality aluminium one can also work very well.
Your looking for features that isolate vibrations (reduce the hum/rattle) and cases with good cooling paths (minimise fan speed = reduced noise)
The stock 4870 cooler will be the noisiest item - think about an aftermarket cooler maybe
Next will be the stock E8500 HSF at load - AC 7 pro is a inexpensive quiet option.
(giving some OC'ing cooling for the future)
Stock case fans would most likely be next - use speedfan/bios/other to get them down to 7v or replace with alternatives (visit the cooling section for details as I don't use fans any more

Decuple your HHD - bungee mod/but rubber mounts.
Noisey psu fan can be a problem, but the earthwatts in the sonataIII is quiet good.
- SonataIII
Steel (check)
Controllable case fans (check) Triccol fans are ok on low speed noisy at high
Quiet psu (check) also 80+ so efficient too
Rubber HHD mounts (check) bungee mod is better but a good start
Air flow (reasonable) - many cases are better, but at stock timings not an issue
Drive bay door - does have some acoustic advantage as contains system noise
With the included power supply it good value for money, not an OC'ing case but a millions times better that a office pc case.