Our 8 year old kenmore dryer won't do anything when you press the start button. We don't hear even a hum. It worked one day and then just wouldn't start the next day. We checked the fuse box and everything is fine.

Here is a good link for basic troubleshooting... But 10 to one the culprit is a thermal fuse or thermal cut off.... they are usually pretty cheap and easy to replace....

http://www.partselect.com/dryer+dryer-wont-start+repair.htm

One final note on these: if you find your thermal fuse\cut off (or element) open, before replacing it, be sure to check the airflow out your dryer's vent system. Even though these devices do 'wear out' with age, poor airflow's the #1 cause of failure of these devices, and if your vent's partially clogged, a new thermal won't last very long, but will do its job again, and leave you 'cold' (a new element won't last long if the vent's clogged, either).

More info and pics:

http://www.davesrepair.com/DIYhelp/DIYthermalfuses.htm