I've used an electric mower (Black and Decker as a matter of fact) for about 5 years, and actually, nothing happens when you run over the cord, but I wouldn't suggest making a habit of it.
The cord slips down into the grass and the mower height has always prevented any problems. But, one of the many great things about electric mowers is that you can stop AND start the blade simply by squeezing the handle. So, let go of the bar and run over the cord and then squeeze again - all problems avoided.
I'd encourage anyone not in the "riding mower required due to size" category to look into an electric mower. They are maintenance free and you throw away the gas and oil cans you keep in the garage.