Here are your parts, includes a wiring diagram.
Replacement parts for MAYTAG MDE3706AYW DRYER- ELE | AppliancePartsPros.comThe cycling thermostat should be 5 connector, see Item 27 in Section 1.
If you click on the picture a new page will open if you look closely at the back of it you will see the 5 connectors.
For the thermostat contacts it has a common, a normally open and a normally closed, shown as the CYCLING THERMOSTAT on the wiring diagram..
It also has 2 for an internal heater, shown as THERMOSTAT HEATER on the wiring diagram.
The normally closed contacts should be closed (0 ohms) at room temperature.
Have you checked the power to the unit. The motor runs off of half he line (120 volts) but the heater requires the full 240 volts.
Try flipping the breaker off/on slowly a couple times.
Sometimes you can loose half the line without actually tripping the breaker.
If this does nothing, check the voltage at the plug
L1 to L2 should be 240 volts
L1 to Neutral and L2 to Neutral, both should be 120 volts.
If OK
Unplug the unit and check the wires at the terminal strip in the machine to make sure none are loose or burned out
If OK
Check the power at the terminal strip.
Do this with the heater off and on.
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Be careful as 240 volts is lethal !!![/COLOR]