My kenmore HD60 series dryer drum does not turn.

ALWAYS UNPLUG YOUR DRYER BEFORE ATTEMPTING TO DO ANY WORK ON IT.
Sounds like your problem may be the belt may have slipped off of the motor pulley, or idler arm pulley.
If you have a front loading machine.
Remove the electrical console that holds the timer switch, start switch, etc. from the top of the dryer. Be real careful not to knock any of the wires off of the timer switch. Then remove the top of the dryer.
Remove the front of the machine, this will expose the belt and drum, remove the drum and belt and this will expose the drum rollers, idler arm bracket with pulley, and motor with pulley. Pay attention to the route of the belt, this will come in handy when you go to reinstall the belt and drum.
Sometimes the belt will slip off of the idler arm or motor causing the drum not to turn.
While you have the drum out, I would recommend that you lubricate the drum rollers at the shaft of the roller with a good grade of machine oil like 3 in One oil. This will save you a problem later with the rollers getting noisy, also check your drum rollers for flat spots, if they have flat spots on them this will cause a noise like a thumping type noise. You will need to replace the drum rollers if they have flat spots on them.
I will include these two websites for parts and further help:
http://www.partselect.com/
http://www.appliancepartspros.com/index.aspx
Both also have chat assistance 24/7.
Good luck,
I hope this helps