Wolf's Rain, I'll start by saying isn't a cyberpunk. It's not a departure from that atmosphere though, and it does have a similar atmosphere in many ways. Technically it's in a futuristic setting, although that isn't really a huge part of it. It might be worth a look though, as it's an interesting show.

Darker Than Black is a personal favorite. It's a rewarding show to watch, as it has fantastic action sequences and is pretty solid in other respects. There's also its little brother Phantom: Requiem for the Phantom which is more or less DTB except with semi-normal hitmen that use actual guns and bullets.

Gunslinger Girl might be a good fit too. It lacks some of that mystery element to the GitS atmosphere, but I consider it to be very dark. It has an almost eerie contrast between light-hearted and mature content because of its plot and characters. It's also about hitmen, so it has that aspect as well. It technically has cyborgs as well, but... not really the cyberpunk themes. This is another anime I very much enjoy.

Hell Girl could be of potential interest as well. No action really, or sci-fi themes, or anything of the sort really. More supernatural with some slight horror elements and a lot of mystery and intrigue. It's a very darkly themed show though and revolves heavily around death. This one's a bit of a stretch, but it has that mystery and darkness that I feel heavily contributes to GitS. A good show though, but I'll leave that up to you if you so choose.

Eden of the East will be my last mention. Yet again, not really possessing of action, or even murder, or any real sci-fi anything. However, heavily dipped in mystery and a very intriguing show to watch. It does have some darker themes but it's a bit subdued in that way, and has some lighter scenes. It's a very, very loved show though. I do liken it to that feeling of watching the mystery get solved in GitS, so that's why I'm mentioning it.