so, why didn't that garage take a look at your mounts? that's probably what it is, pop the hood and have someone look as you put it in gear with your foot on the brakes. typically you can see the mount in question rock back and fourth. strut mounts are possible, but not likely.

 

 

I'm thinking it's your control arms, part of the suspension on the passenger side where if you have leaks, especially power steering.. It can leak onto it and it causes problems.

 

This seems like what a failed mount would do.

 

+1

 

If it was one of thoose windup and get go like an turbo lag spooling up but wait before you launch. It will feel like as if you were rear ended engine revs up and than all of a sudden the car starts moving again. This is trans issue, If your taking off or coming to an stop a hard thud with lot's of viberation, as if you were hit by a car this would sound more like an engine mount issues. The trans can do that the same but it would most likely be a torque converter issue. The first thing you can do is pop the hood and check with the car running select the R and D gears with the foot on the brake pedal. In each gear one foot on the brake and the other on the gas, rev the car and check and see the engine rock back and forth. If the car does rock back and forth the problem has been diagnosed. The side of The engine that moves violently needs, it's mounts changed. The engine should not move at all, very slight and minimum moving is normal.