As cheshirecat said, I think it depends on which weed you're working on. I like Anne's bread knife method for the perennial grasses in my groundcover where I just can't get the roots out anyway. Cut the tops off enough times, the roots will eventually expire.
IF there's a chance to pull the roots out - small plants or shallow rooted annuals - it's just about the same amount of work as cutting off the tops anyway, isn't it? Plus you increase your odds of eliminating that weed in case it does regenerate from root cuttings.
If the weeds are in an open area like a mulch pathway, burning the tops under clear plastic in the sun has worked well, as long as it doesn't rain soon after . . or your irrigation is sprinkling where it shouldn't be.
