Mine is fairly simple.
Your garlic choice and the amount is pretty important. I made two batches. One with fresh garlic and one with jarred garlic in olive oil.
Fresh raw pressed garlic (from a head) in yogurt can be overpowering a day later. If you use minced raw garlic in olive oil it will be tamed down a bit and stable days later. It's the kind in a jar you'd find in the produce section of your grocery store.
Funny you mention this because I stopped using fresh garlic for the most part due to the heat. I don't mind it but my wife won't stfu about it. The jarred stuff is pasteurized which kills the enzymes that make the garlic hot. That stuff is a little mild on flavor but has 0 heat.
I have found a slight workaround though (i think from serious eats). If you crush your garlic directly into an acid, such as lemon juice, you'll denature the enzymes that make garlic hot. I've done this quite a few times and i think there's real merit to the method. Works really well with hummus.
Anyways, i think the 2nd key to the sauce, besides real greek yogurt, is a very very modest dose of garlic. Restraint is absolutely crucial. If using fresh garlic the ratio is like a clove of fresh garlic to a cup of yogurt.
I skip on the dill personally. I had a bad experience one time with a leg of lamb rolled up with feta and dill, and topped off with about 8 irish car bombs. Of course i puked, and all i could taste was dill. Haven't had dill since. Had many more irish car bombs though. Never a good decision.