Hi April,
As you've already thought, I have to agree that two brand new dryers in a row with exactly the same defect being wrong with them seems at best... unlikely. So let's try and figure out how things work and perhaps that will lead us to some sort of clue as to what's not working.
We'll start out with the electrical outlet. There are three prongs (at least) on your plug; two "hot" prongs and one "return" or "ground" prong. Now each of the hot prongs, when measured against the ground, will read 110 VAC. In order to get the 220 VAC that the dryer's heating element wants, you go from one hot prong to the other, so 110V + 110V = 220V.
The thing is, if one of those two prongs isn't working, then it's entirely possible to get what you're seeing; the motor and most of the other electronics and motors in the dryer will work fine because they run on 110V, but since the heating element (the only thing that uses the full 220V) isn't getting power, the dryer won't dry.
Now how can that happen? The most likely possibility is that one of the two 110V circuit breakers feeding the dryer has tripped, leaving the other one active. That would be my personal bet, but that's tempered by the fact that if that's what happened, a professional electrician should have easily caught that. Regardless, go to your circuit breaker panel and check; look for the one marked Dryer and verify that both breakers are all the way on, not in a halfway or off position. If not, switch it to the off position then back to on.
Okay, you'll need a voltmeter for the next parts. First, set the meter to measure AC volts, and if it has different ranges, set it to read at least say 300 volts; that'll cover the 110 and 220 volts we're expecting. Now put one probe in each of the two side prongs and you should see 220V. Move one of the probes to the middle and you should get 110V. Swap it around and connect the other side to the middle and again you should see 110V. If one of the 110V sides is missing, you've got a wiring problem between the circuit panel and the outlet.
Assuming you've got good voltages at the outlet, next step is to make sure you've got that same power getting to the dryer itself. There is usually a panel covering the place where the dryer plug connects up to the dryer itself, so open that up so you can see the three connections corresponding to the three plugs in the outlet. Plug it in and go back through the same measurements you just did, on the dryer this time. You should get 110V going from either side to the middle, and 220V connecting across the two outside terminals. If you've got power at the outlet but not at the dryer, then you've got a bad cord; if you used the same cord on both dryers that would explain it.
As I said, a professional electrician should have caught any of the issues I've described, but if you've got a voltmeter you can do those checks yourself. You never know, there are people who call themselves professionals who I wouldn't want changing a light bulb so it's not impossible that your electrician missed something.
Those are my thoughts; take them for what you paid for them, but I hope they help.