I read all the comments/advice in answer to this problem.
Here is what I did to fix the problem (I did everything on this list at the same time, so I can't tell you which item is the one that actually worked:
1. I switched back to the egg-shaped earbuds that came with the iPod Nano (I had been using the old pancake flat ones that came with the old iPod Classic/video player as they are more comfortable to me)
2. I turned off the voiceover in all of the following locations:
a. Settings, General, Accessibility, Voiceover - turn to off. Also, turn off "speak hints".
b. Fitness > click the "i" icon > Run > Spoken Feedback > change to off (note: this was set to On on my player)
c. iTunes - bottom of Summary section, Accessibility, uncheck the Voice over checkbox