It's not the gas tank. I have the same vehicle (2008 V6) with the same issue. It comes and goes and it's usually only an issue when the vehicle is cold - rarely does it do it when it's hot. I've actually had mine stall several times. There are no codes either, which is what makes it weird.
I've narrowed it down to a couple of things
1 - one of the fuel pumps
2 - torque converter in the transmission
3 - the way the secondary air injection system works on startup
Of the three, I think #1 is most likely. Based on what I've read and been able to find, of the two fuel pumps in the tank, one is used for startup and the other is used for supply to the third (high pressure) fuel pump at that engine. But, it's unclear which fuel pump is on each side of the car - I've looked at the technical docs, and all sorts of things. Rather than throw money at it chasing a solution, I've just chosen to live with it.
Now, it could be the torque converter in the tranny too - or the valve box for that matter. But since it only exhibits symptoms at low/no speed, I discount this. if it was the torque converter, it would be slipping at higher rpms, not around idle. If it was the valve box, I would expect it to exhibit other symptoms (banging downshifts, etc). But it isn't.
I can tell you that the problem has VERY slowly gotten worse - but if I think back, I would occasionally get some slight off idle hesitation when the car was new to me (at 35K miles). It now has 95k. In between it's had it's valve box replaced.
I wish I had an easy answer for you - often times I give it a rev to about 3500 rpms after startup and before I move and that seems to eliminate the issue. It's not going to break. It's annoying as hell though. I hope that helps - you aren't alone in your issue.
As for my own issue, I've been looking for a '13 Touareg TDI Exec or Q7 to replace mine. I love it to death - it's been a positively awesome vehicle. But I just can't stand the mileage any longer. And when it goes, so will this problem!