When the engine mounts of an automobile go bad, the engine could actually sag as much as a few inches--which may cause shifting linkages to not work properly. With the engine drooping, the transmission is also likely affected. Therefore, if you notice that it's difficult to get into or out of certain gears, or the transmission isn't shifting properly, a bad engine mount could be the cause. Once the broken motor mount is replaced, inspect to see that no linkages actually were bent while the engine was drooping because the automobile will likely still have symptoms if this is the case.
On older vehicles and newer trucks with a radiator fan that is connected directly to the engine, a broken motor mount could cause the engine to lift or shift slightly under acceleration. This can cause the fan to make contact with the radiator. V8 engines are most at risk for this because of the radiator fan placement and because of the torque they create, which can lift an engine several inches under hard acceleration. If you notice that the fan is touching the radiator in any way, inspect to see that the motor mounts and radiator mounts aren't broken, and fix them immediately if they are.
Sometimes when a vehicle has a broken motor mount, the engine is allowed to lift during acceleration, which may affect throttle cables or linkages. If the throttle sticks wide open under hard acceleration or towing, inspect the motor mounts and throttle linkages immediately. For obvious reasons, any automobile on the road with a sticking throttle can be very dangerous. Again, if the problem persists after new motor mounts are installed, inspect the linkage for bent parts and repair or replace them.
Some manufacturers have provisions to make solid engine mounts for their vehicles. Normally, a motor mount is made of two pieces of metal with a hard rubber insert in between. The metal on either side is the attachment point for the mount, while the rubber insert absorbs vibration and allows for some flex so that the passengers can't feel the engine working. However, if you experience broken motor mounts often, check to see if solid mounts are available for your vehicle. There will be some slight vibration felt inside the vehicle, especially upon takeoff, but the solid mounts are much sturdier units and will last noticeably longer.
