Well, dig sites can't appear unless you've looted ruins or had a fight, so if you never let any enemies get within three tiles of your cities, you won't be building many landmarks. Having said that, there's a set number of total dig sites that the game generates, so waging repeated battles over the same patch of land won't leave you with an entire field full of artifacts.
Also, I don't think dig sites can pop up on a tile that has a resource in it, so you might want to pay a little attention to where you kill that barbarian that is harassing your city.