Like others have said, it's random which villagers ping you and ignoring them won't help. Actually, I've been trying to move out a villager for the longest time and I kept forgetting to talk to her because her house is so far away from everything, until she ended up pinging me yesterday.
The method I use for getting rid of villagers I don't want seems complicated but it's pretty simple and safe. I'll outline it below for you, there was a thread with some variations on it but I can't remember where it went or what it was called but it's where I got this idea from. If you don't like TTing, there aren't any specific methods you can use though, other than just being patient and waiting for the villager in question to ping you.
The method I use is below. You'll need to have an empty slot for a new character (don't worry, it will be free by the time you're finished).
First you'll have to pick a week that has no events in it and the Sunday that starts that week (in the game, weeks start on Sundays). The example I'm going to use is the time I did this that sticks out the most for me, when I picked June 1 2014.
Use the 3DS calendar and set it to that date. NOTE: If this is more than a day, maybe two tops, for you, do NOT do this yet. Wait for a villager to ping you to move, ask them to stay (unless it's the one you wanted to move out to begin with) and only then set the calendar to the date you've chosen.
It should now be that date in your game (if it isn't and your game is ahead or behind, you'll have to calculate which date to set your 3DS calendar so that it is the nearest Sunday beginning an eventless week in the game). Load the game as your mayor, then immediately save and quit.
Go back to the 3DS calendar and set it one year ahead. In my example, instead of it being June 1 2014, it should now be June 1 2015.
Go to the game but do NOT load as your mayor. Start as a new character. The answers you give to Rover don't matter since you won't have to keep this character. Once you arrive at the station, Porter will give you a town map and tell you to go to the town hall. Do NOT go to the town hall. Instead, check to see if any houses are missing. If nobody's missing from your map and you want to be extra careful, go visit the animals who are at home and see if they are in boxes (you don't need to go to the town hall for this, so don't) but that's usually not necessary.
Quit without saving. Go back to the 3DS clock and set it back one year. In my example, it would now be June 1 2014 again.
Load the game as your mayor and go find the villager who was missing when you visited the future. If they don't ping you, talk to them like half a dozen times and see if they ask you to move. If they don't, save, quit, then reload (still on the same date) and try again. Keep doing this step this until they mention moving, usually it takes only 1-2 tries.
If they're someone you want to keep, ask them to stay and save then quit. Go to your 3DS calendar and set it one year ahead, so it would now be June 1 2015. Load as your mayor, immediately save and quit. Go back to the 3DS calendar, set it another year ahead (June 1 2016 in my example) then start as a new character to check the map as described before. Quit without saving, set the calendar back one year (June 1 2015) then use your mayor to talk to the missing character. Keep doing this until it's the villager you want to move out. The next time, you'd start with your mayor on June 1 2016, use a new character on June 1 2017, then back with your mayor to June 1 2016. This can take you some years into the future because you're always going forward one year with your mayor, then forward another one with the new character, then back one with your mayor. Don't worry.
After the villager you want to get rid of asks you to move, tell them to go using your mayor and save and quit. Then go to the 3DS calendar and set it to the original date. Because going back any amount of time counts as one day, you'll have to calculate your villager's new moving date. Using my example, if they said they were moving on June 6 that would be five days, so they would now move in 4, meaning that if your original date is May 28 they'll now move on June 2.
The only downside to this method is that you get a lot of weeds and also bed hair (you can fix the latter by going to shampoodle). If you have Beautiful Town as your ordinance, your flowers won't die. There's other variants like going week for week or month for month but I find this easier to keep track of. It sounds complicated but you'll get the hang of it, you just have to be careful to not load as your mayor when you're going ahead.
The June thing was an example, you can use any other dates.
