Procedural mesh with texture atlas, how to get rid of material artifacts?

I am basically making a minecraft clone and everything is working great except the material. I just cannot figure out a way to get rid of the artifacts that appear when I render my chunk mesh.

I am currently using the "standard" built-in shader with the "cutout" rendering with the alpha cutoff set to a very low value but it doesn't really mitigate the issue. Do I have to write a custom shader for this or maybe I need to do something else?