Stolon and rhizomes: to pull or not to pull

I've been pulling some of the stolons or rhizomes from the rows of berries I have. Sometimes it seems like a good idea. The 'string' is right near the surface of my deep mulch, comes right out with no obvious bits left in the soil, and it's kinda' fun. But I have read that such pulling is just propagating more starts for the invasive species as there are ALWAYS bits and pieces of the stolon/rhizome left in the soil, and that the best practice is to just pull the green tops of these plants and they will eventually starve without such tops. What is the experience/consensus of those who have had to deal with this, sans chemicals, which I will not use?

I've recently been pulling a nasty creeper spreading through my mulched beds. In the beds with mulch i've been able to pull whole string before they have rooted properly - very effective. In my other beds, without mulch, pulling just snaps them off at the roots and they regrow. I would say that if your mulch is nice and deep and they pull cleanly, then go for it.