dandies are quite deep rooted so really you need to dig right down to make sure you get the lot out or they will just grow again. I do use an organic weedkiller but mainly on weeds growing in paths etc.
Boiling water right out of the kettle will kill most weeds, again best for patios and paving,
The other method I have used is to suppress the light source. EG, on my gravel patio if a weed pops up I just move a potplant on top of it for a few weeks or on the lawn pop a stone on top. Grass recovers very quickley.
Also worth bearing in mind is that some beneficial insects use dandelions as a food source.
Hope this helps.