Basically, growing large single crystals is more an art than a science. Once they are 1 mm in size, pure and clear, they are adequate for x-ray crystallography. The several cm sized ones suitable for display take even more than science or art -- they take luck. However, some things to help
-- You have to try to grow slowly. If a warm saturated solution grows many crystals when taken to room temp, then a hot saturated solution may grow bigger crystals when cooled only to warm.
-- You will have to experiment with different strings, nylon fishing line is a good choice, assuming such things still exist. Whatever you choose, you will still end up with a string embedded in your crystal.
-- If you can get crystals to grow in some sort of gel, you will have a string-less, 3 dimensional "crucible", that slows diffusion, and resists temperature changes. That's a little bit hard to swing however.