How many villagers are you trying to keep? The more you are willing to cycle through, the faster you'll get through 16. You need at least 2 spots to cycle, because it's so hard to budge the last villager in. I'd recommend you have at least 3 spots or it will take forever. You should always have 10 villagers, because I think villagers ping much more frequently to move out when you have 10. So invite them from the campsite, or ask some cyclers if you can adopt their unwanted villagers to cycle through.
People are recommending the foolproof cycling method to you (this is the 2 year 5 day method), but even though I love it for cycling I'd be terrified to use it in a town with villagers I want to keep. If you are willing to let everyone go, however, it will let you cycle 16 in less than a day. There are modifications for people who want to keep some villagers, and I think this must be what Maruchan is referring too.
