If you really want them gone, two treatments of 2,4-D ("Weed-B-Gon" is one trade name containing it), spaced a few weeks apart is nearly totally effective. Dandelions are one of the easiest weeds to control with chemicals that don't harm the grass next to it.
If you want a dandelion killer you can feel better about, 20% acetic acid ("vinegar, but 4x as strong") applied as a spot treatment will also kill them. (As will glyphosate, of course, but if you aren't willing to apply 2,4-D, I'm guessing glyphosate is right out... ;) )
If you're over-run, forget any mechanical methods. Too time-consuming, and if you don't get the whole root, they just come back.
Or, as other posters suggest, re-frame your opinion of them as "not weed" (and hope that your neighbors agree, because you'll be dandelion farming seeds that they'll have to deal with...)