My experience is with a 2005 V-6.
The liquid filled mount on the passenger side leaks and causes shaking similar to what you describe.
Also noise but that built up gradually so I could only appreciate the silence with a new mount.
A test is to have someone rev the engine in park while you observe the engine.
If it tilts appreciably when revved, it points to the mount.
There are other tests for the V-6 mount but replacement was cheap enough and easier than exhaustive testing.
Here's a YouTube testing a I4: