1. You Must have a swimsuit.
2. Swim to the barrier of the water that you can swim in.
3. Wait there for about 5 mins (Jelly sting warning)
4. Get out the water, and run around your town. Some villigers should ping.
Ah, fun. I'm currently cycling for Chief, so only moving out specific villagers. I just TT day by day and talk to everyone. I'll talk to a villager that I'm close with many times, to see if they mention anyone moving. And make sure I walk in front of everyone to see if they ping. Usually every couple of days (sometimes longer) someone will ask to move out. It takes a while for it to be someone I want to move out, so it is tedious. But there's really no quicker way imo that is as safe.
Basically I use the same type of routine that FireNinja linked to. And if someone has just asked to move, it's pretty safe to TT two days since it's unlikely that you'll get two requests back to back. The only reason I don't is because I'm collecting bush starts, so I buy them every day.
So all I can say is make sure you talk to the villager you want to move out (friendship works, ignoring them does not), and keep trying. Sometimes it takes a while!
The system I explained is for that purpose, so you don't lose the villagers you want to keep. If they ping and ask to move, tell them to stay and continue on. If you hear they are moving from another villager, keep walking in front of them and try to get them to ping so you can tell them to stay. If they are inside, TT a few hours so they are outside. If someone you want to move pings and asks, tell them to go (and if they ping again before they are in boxes - ignore them because they may decide to stay).
I am currently cycling villagers out and I have 7 villagers I want to keep. So it can take quite a while before one of the others I want to move asks. With just 3 villagers you want to keep, it should be a lot easier. c:
edit; The link FireNinja posted explains this much better than I, so I would recommend following that.