Factory Mazda engine mounting leaves a lot to be desired, the idea is that the stock mounts were designed to prevent vibrations from entering the cabin. The issue is that the rear mount is so soft that it does not support the engine effectively during shifts and hard acceleration.
This can be fixed with an upgraded rear engine mount, there are many on the market with varying stiffness or "Durometer", the general rule is that the stiffer mounts will cause more vibration but will hold up to the most aggressive driving. While softer Durometer mounts will absorb vibration better and will work well for most street applications, but may not hold up to extreme driving or motorsports applications.
These mounts are available from many vendors in may different levels of stiffness to meet your needs, including one from Cobb Tuning!
Damond Motorsports, James Barone Racing, and AWR are some other options for upgraded Mazdaspeed engine mounts.