This is by far the most common dryer problem of all-time. It's usually a bad glow-bar or ignition cable, depending upon the type of ignition system that you have.

The glow-bar system sends electric current through a heating type of element until it glows hot enough to light the gas (this is used on most older style dryers) An I.R. sensor (infrared sensor) determines when the glow-bar is hot enough to light the gas, it then sends a signal to the gas valve to turn on.

To test this system, open the front cover where the burners are; start the dryer in the heat cycle; you should see the glow bar getting red hot (the glow-bar sits inside the burner box just above the burner tubes - there are two types; a round one that has spirals, and a flat one that looks like half of a coat hanger except smaller. If you have a hard time identifying them go look at the pictures of them in the store section) If nothing glows you will need to take a voltage reading across the glow-bar leads (there are 2) as you first start the dryer.  No voltage means it's something else.  Voltage means replace the bar (a replacement glow-bar should be kept in stock). 

If the glow-bar lights and does not go off then the I.R. Sensor has failed.  (I usually replace the glow-bar along with the I.R. sensor because one will take the other out, and it's just good maint. procedure to do as such.)

If the glow-bar lights for about a minute then goes off and you do not hear a click sound come from the gas valve (this click sound is the coil in the gas valve opening) then it's probably a bad gas valve coil (see below).  If the glow-bar lights, then turns off (in about a minute) and the gas valve clicks on (and you can hear and/or smell that the gas valve is open) the burner tube holes are plugged and/or the glow-bar needs to be adjusted closer to the burner tube.  To clean the burner tubes, you can either shoot compressed air into them, or remove them and run water through them.  If it's an older dryer it's best to remove them and run water through them so as to get all the compressed lint completely out.

If you have a spark ignition (looks like a spark plug wire from a car...in fact it is.) The way that this system works is, instead of lighting the gas with a glow-bar, it lights it with a spark.  To test the system, open the burner cover, turn the dryer on in heat mode, you should hear the gas valve click, and the spark igniter sparking, (in fact, it may even light the gas, but then goes right back off.)  If this is the case then most likely the ignition cable is bad.  Most people think that just because they can hear the spark, and see the gas light that the cable is OK, however, the ignition module sends a small amount of voltage the opposite direction through ground, through the fire, and through the high voltage ignition cable back to the ignition unit to let it know that the gas has lit. If the ignition cable has a even the slightest crack in the carbon core cable; that small amount of voltage cannot jump that crack like the higher voltage.... So replace the cable. (if it's a weekend and your faced with a downed dryer; either buy one from an auto parts store, or borrow one from someone's car...)  The other thing that it may be is plugged burner tubes, if it looks like there is a lot of lint in the burner compartment you may want to try blowing out the burner tubes or removing them and rinsing them out with water.

If you do not hear the gas valve click (as mentioned above) and you do have spark or the glow-bar is cycling on and off, then odds are that the coil on the gas valve is bad (most gas valves have 2 coils as a redundant safety feature) in all my years of service I've only replaced a couple of entire gas valves, it's always been the coils that sits on top of it. The coil can be replaced as a unit (4 -screws hold it on) it usually has 2 or 4 wires going to it.  (It's rec. that this be one of the parts you keep on hand because alot of times repair techs get back ordered on them).  Be sure to shut off power and gas before replaced and mark your wires.

The other common problem associated with a no heat condition is when the sail switch fails to pull in.  The sail switch pulls in when the dryer is started via the air circulation.  If the circulation gets reduced either from heavy lint build up on the lint filter or a blocked exhaust vent the sail switch won't close all the way and the heat circuit will shut down. (I use to work in San Francisco, CA and the homeless people would dry their clothes over the exhaust vents on the roof or in the alleys which caused these sail switches to open and shut off the heat circuit...i.e. service call) Another thing that happens alot is the maids would either clean behind the dryers or store stuff behind them knocking the sail switch out of alignment without knowing it (alot of sail switches are behind the dryer about a foot above the ground).  There is an adjustable weight on the sail switch, the proper way to adjust it is: with the dryer off, move the weight just enough so that the slightest movement will cause the switch to click on (switch should not be tripped when dryer is off).  Test it by running the dryer, the air circulation should pull it closed.

For years American Dryers put these high limit switches in their machines that had reset buttons on them (look inside the lint compartment - just under the basket) If the dryer got too hot the switch would trip.  If this is the case be sure to check the temp. (I would always replace the temp. sensor whenever there was any doubt just to be safe.)

The other thing with American and Cissell dryers was that the connections would always come loose inside the lint compartment (housekeeping staff would knock them loose actually when they would go to clean them) so be sure to check them.

Thermodisks are heat sensors that are about the size of a nickel or quarter with two screws. They are placed in lint compartments, burner boxes, and exhaust vents. When they heat up to a preset temp. they will open up (electricity won't flow through them) when the dryer is cold you can check them with a continuity meter (turn power off).