We recentley ditched the dummy with DS2. He gets stimulated very easily so we used a CC approach. It wasn't to get him to sleep through neccessarily, but just to be able to settle without the dummy.
First night we put him down as usual and he cried for about 45mins. Was awful. I stayed in the room for the first half but was worried i was actually antagonising him more so started going in every 3 mins till he went off. He then woke a couple of times in the night and I fed him, as each time it had been more than 4 hours so wanted to make sure he wasn't hungry.
We did the same for naps too.
It took longer than I thought, although after the first night he only cried for a few mins, he was still doing this 2 weeks later - and sometimes waking and screaming the house down only 45mins after he'd gone down. So we've tweaked it a bit and if he is really screaming when I try to put him down I will feed him to sleep. If he is just grumbling then I give him 10mins and he will either drop off, or it will build to a scream so I will feed him.
I also have a toddler to look after, so DS2 is invariably overtired when I manage to get him down for a nap, so it is unreasonable to expect him to be able to settle himself then. But without the dummy now we have pretty much eliminated the 45min nap thing, which used to drive me mental! Now, if he naps for only 45mins I assume that is because that is all he needed and will get him up.
As i said we did it to get rid of the dummy not to get him sleeping through the night, and he now sleeps from approx 7pm to 8am with maybe one waking in the evening (still don't know why he does that!) and then once in the middle of the night. So much better!
Good luck - keep posting to let us know how you get on.