Credit for this answer goes to my mum.
When we (wife and I) moved into our current home 4 years ago, the front and back lawns were covered in dandelions (and other broadleaf weeds). I didn't want to go the herbicide route, and after speaking with my mum, she said the only way to truly get rid of (control) dandelions is to hand remove them (important: you need to remove root n' all) and recommended using a small hand garden trowel for doing so.
I didn't have a "small hand garden trowel", instead I bought a small (and cheapish) builders trowel and used that for 3 years:
Then last year on clearance (paid $1) I picked up a Fiskars Softouch Weeder:
Both of the above hand-tools have worked well in removing dandelions (roots included).
The first three lawn cutting seasons I pulled a lot! of dandelions, at least once a week I would walk the lawns and hand remove all the unwanted plants I saw (all of them went into a plastic bag for offsite disposal).
During last year's lawn cutting season I noticed a big drop-off in the number of unwanted plants I was removing, except for crabgrass (but that's a whole other story).
This years lawn cutting season I've only had to remove a handful of "small" (young) dandelions, this is to be expected as I can't control dandelion seed heads blowing into my garden (though I wish I could).
Yes, getting any unwanted plant under control via an "organic" approach is going to take patience and time, but if you stick with it, you will see (excellent) results.
And bare in mind, even if you go a "non-organic" route, you're pretty much "forced" to continually use the chosen method if you don't want to see the unwanted plant(s) return.
If you wish to read more about how I approach "organic" lawn care, go here on SE:
Good luck!
