Talk to them more works better than ignoring!
I read somewhere that there's a game "ladder" when it comes to friendships. The top of the ladder is when they give you a picture of them. The only way to reach the top is if you talk to them often.
Here's the catch- The game doesn't want you to reach the top so easily, so the closer you get to a villager, the faster it is to make them want to move. (So it'll be harder for you to get a picture.)
If you choose stay, you'll be able to reach the top and obtain a picture of said villager, but otherwise, if you ask them to move, they will.
Ignoring them, however, will put them at the base of the ladder, and will take much longer for them to move. If they've just moved in, you can introduce yourself then ignore them til they move, but that might take a week or so.
I hope this clears anything up! This worked with me with some of my villagers so I hope it works for you. ^^