Check the condition of the motor and trans mounts, as well as the exhaust system.

Worn or broken mounts might be causing the exhaust to bind or make contact with something else. They are easy to check: with the hood open, brake torque the engine and see if it moves around excessively.

Rotted exhaust piping, broken hangers, etc, may do the same. A good visual inspection and shake down will reveal any problems.
2011 Crown Victoria LX - purchased @ 37k miles
currently @ 100k miles
Dark Toreador Red Metallic exterior
Medium Light Stone interior