I don't know if could run without music. I guess one has to be sensible re: isolation from the surroundings (traffic/muggers). I never found this to be a problem as the phones I use are not that isolating to start with plus I do not max out the volume and there is always the option to stop the music temporarily when running through some more alert-demanding areas.
Recently I got some Meelec M6 which are great for sports, the shirtclip, over-the-ear style and the bit that one can tighten the cables with under the chin help to keep the phones in place and there is no cable noise. I also had no probs with the EP630 (over the ears style) where I ran the cable inside my shirt - just remember to rinse the cable afterwards as the sweat can eat up the cabling....I never found that I sweat around my ears so I do not worry about the phones themselves.
Just do it!
