I've found Tourna Grip to be fantastic overgrip. They are not very tacky, but they perform their best when you start sweating and the sweat starts running into your hands and grip. They don't last very long for me though; 1-2 long sets before it starts fraying/peeling. But I just flip over it and re-wrap using the other side of the overgrip -- good for another set or long hitting session.
There's a reason why so many pros have used this overgrip for a long time.
I've since been using Solinco Heaven
overgrip because I think it improves on Tourna Grip in almost every way. They both have similar feel (non-tacky, yet slip resistant), but Solinco Heaven lasts a bit longer, is slightly thicker, and is just as absorbent.