I had a similar idea to pitspotter, with visible instead of audible cues.
(I hope this hasn't been suggested before, but...)
My idea was that the edge of the screen would glow red (or whatever colour) if there was a car in that direction nearby. For example, the bottom of the screen would glow if there was a car close behind you, or the left side of the screen would glow if there was a car on your left side.
The intensity of the glow could also be used indicate how close the car is to you.
I think that this would adequately simulate peripheral vision as it would give you the "feeling" of a car being there and, in effect, it would give you an extra "sense", but you would still have to turn your head to see exactly where the car was. Overall, I think it would be a valuable addition without being too arcadey.