because you asked

I'm curious, Dave, what do you do about weeds in your garden? I haven't been going organic like you, partially since I'm not growing much of anything that I'm going to eat, so I got a bottle of weed killer and sprayed it rather heavily over a corner of my lawn that was grown over with clover and dandelion. It had pretty much zero effect.

We've got a bumper crop of dandelions ourselves. The only way to remove a dandelion is to pull out its entire taproot. If you just break off the top of the plant, it'll grow back.

There's a couple ways you can get of them with varying degrees of ease:

1) Pile on the mulch. The dandelions will grow through, but because the mulch is loose, they won't have a very good grip and will come out more easily when you pull them. If you don't want to pile mulch on, try compost.
2) Mix equal parts vinegar with water and pour the mixture on the offending spot. This kills weeds and other plants since the vinegar burns the plant from the inside. I've never used it because I'm not sure how the vinegar would affect other plants and grubs, though I'm sure it's a lot less nasty for the environment than conventional herbicides and lawn chemicals.
3) If you have a particular part of the lawn in mind, dig it up or rototill it. The rototiller especially will destroy any plant in its path, clearing the way for you to plant whatever you want.
4) Enjoy them. Personally, I love dandelions. They look nice, their seeds are fun to blow when they go all white and puffy, and if you collect their leaves early enough, they make a nice addition to your salad. (Let 'em go too long and they get very bitter.)

If you want to prevent dandelions, it's too late at this point, but next spring when your crocuses -- or the crocuses of somebody else who lives relatively close, at least in the same county -- start to bloom, hie unto your local Agway store or the equivelant and buy yourself some corn gluten.

Corn gluten is a byproduct of corn processing. It has no effect on established root systems like your grass, but when you spread it on your lawn, it acts to inhibit new seeds from germinating. Put that on your lawn at the start of the spring, and it will greatly reduce not only your dandelions but your crabgrass as well. Best of all, it biodegrades into nitrogen and fertilizes your grass as well.

Corn gluten is effective at inhibiting seed germination for up to six weeks, although you can transplant established annuals and other plants in the treated soil at any time. (You also can walk on it and let your children play on it, which you can't do with herbicides.) After the six weeks pass, overseed your yard with new grass seeds, and in the course of a couple years, your yard is going to be pretty much weed free.

When I pull a dandelion or other weed out of my garden or flower beds, I toss it into the middle of my compost pile. The heat kills the seeds, and the whole mass decomposes and becomes a part of my garden the next season.