I had the exact same problem on my 2001 3.0 earlier this year. Very rough vibrations when transmission in drive at full stop (would go away in neutral and park, or when driving). Changed the engine mounts and 90% of the vibrations went away.

Changing the torque converter resolved the leftover vibrations, but the reason for changing it was because I also had some shaking when accelerating, so getting rid of the rest of the vibrations could be considered a bonus. :P

Similar issue: Replaced the engine mounts and tranny mount, still vibrated. I did have to replace both front drive shafts and flex discs. Now the vib is very, very slight. However, one mechanic stated it's an idle issue; to low. Than someone stated Torque Converter, but my tran is smooth n sweet. Not sure what it is but I just live with it now. Let us know what you find. BTW, my engine mounts were oil filled and had leaked while sitting for 6 months.

2001 BMW x5 E53 3.0i. Same Idle problem
 

Can you tell me if changing the engine mounts solved your problem? I have checked everything (DISA, ICV, CCV, did Valve Cover gasket, new fuel pump and filter, new belts and idler pulley and a/c pulley)...
It seems to be worse in the cold mornings too.
Any followup or help would be appreciated...Thanks!

I’ve just been living with it. The car has 200k miles and at least a dozen problems. I’ll likely be junking it soon, it is worthless.

John Galt, but did you ever change your motor mounts?

I never did. I still have the vibration.