For weeds in my garden beds, I just pull 'em (and then pull 'em again). We live in a very lush and verdant environment, and this is just part of the price I pay. I do mulch like a crazy person (bark in the perennial beds and loads of shredded leaves in the raised veggie beds), but the mulch inevitably gets pushed around by hard rain or critters.
For weeds in my garden paths, I use pickling vinegar (which has more acetic acid) or I pour boiling water on them. As a Master Gardener, I've been taught to be sensitive about recommending herbicides, but there are a few that are less damaging to the environment than one might think. I've also torched the weeds, which is both fun and permanent.