For the first part of my career, I never missed a day. I had a streak through HS and let it slip a little in University.
Then I would take the occasional day off, but always a very easy day each week...5 miles jog with massage afterwards. That worked great.
Later, I moved to taking a day off every week or so. Sure, you need to adjust your mindset a bit in terms of miles/week, but in the end, you will get more out of the other training days and stay healthier.
Jeff Schiebler (27:36 and 1:01 half) took a day off per week. A lot of the Kenyans do also.
The bottom line is that if you are training moderately hard, run every day. If you are really training at either a high intensity or volume, then a rest day will do you good.