2001 Dodge Neon Weird noise after timing belt replacement

Noises problem
2001 Dodge Neon Automatic 90,000 miles

I have a 2001 Neon with a bunch of minor problems that I am going to get repaired next week (I'm currently out of town on business and not near my regular mechanic). After an oil change, my alternate mechanic discovered that I was in serious need of a new timing belt, so I had that replaced. Once I went to pick up the car I immediately noticed that the noise of the engine when I started it was different, much deeper than normal. I drove it off the lot and discovered that whenever I give the car gas, a deep choppy (but regular) vibrating noise, almost like the sound a VW Bug's engine makes, comes from the left front side of the car. This noise does not happen when the car is in park, even if I gun it up to 3000 rpm; it also doesn't seem to matter how fast I am going, though the noise is quieter and sometimes disappears after I have been driving around for 20 minutes or so. After inspecting it again, the mechanic told me to drive it around for a couple of days and see if the noise is improved before bringing it back in, but I have not noticed much of an improvement, and I am loath to take it back to that mechanic since he suggested it might be a transmission issue (when last week he told me that the transmission was fine).

The problems that the alternate mechanic diagnosed were largely brake problems (I believe I have a cracked rotor), new tires, and a broken engine mount. The car runs fine, and does not seem to have any other problems. The noise only occurs when I am applying pressure to the gas pedal, although when I bring the car to a complete stop, the same noise is made in the instant that the car is coming to rest (and not while it is moving slowly). If my foot is not on the gas, the noise does not occur no matter how fast I'm going (I've had it up to about 65 to test it). Any ideas what this might be? I will have to drive about 300 miles to get back home to my regular mechanic; if it's the engine mount making the noise, will it be safe to return home in its current state? I'd really rather deal with my regular mechanic if I can.