The problem is most likely with the centrifugal switch on the motor.
Under normal conditions, the centrifugal switch operates when the motor
reaches full speed, which is the electrical equivalent of holding the
start button down.

Vacuum the dust and lint away from the motor and look for anything that
may be obstructing the switch. *It's near the point where the wires
connect to the motor.

I've opened the top of the dryer, and taken off the back panel to look
for the centrifugal switch, but cannot see it.
Tilting it on its side-I can't get to it ffrom the bottom either.

When I had the back panel off, I found a troubleshooting guide for the
repair tech. I tried to go through all of the steps, but can't get the
dryer into test mode.
I've tried this many times, it just wont go. If I jump ahead past the
test section of these instructions, the Problem/Cause table says if
"the dryer starts but won't continue" the electronic board assy should
be replaced. I realy wish I could verify the centrifugal switch works
before doing this. I'm sure that cirbuit board will cost close to
$100, if it's even available.
There are some schematics here too, but if I can't get my voltmeter
probes onto the centrifugal switch, it doesn't quite help.
I could use a dissasembly diagram.

We've had this dryer for just over 5 years.