Engine noise

For several months now I get a very loud rumble noise under the hood that sort of sounds like a diesel engine. The truck starts up just fine and runs perfectly. It only happens when I come to a stop or slow down a lot, like pulling into the garage. The minute I give it gas again. It stops. One time I was able to keep the noise going long enough to open the hood to see if I could see anything and the engine was shaking very hard. Again, when I gave it gas, it stopped. It almost sounds like it is going to stall out, but never does. Seems to happen more often the longer I have it running. Like if I am traveling. It’s at the shop now and of course it won’t repeat with our mechanic. He has had it on the computer diagnostic thing and it doesn’t read anything either.
Any ideas? 2008 Ford F 150