I agree that "Za zdorovje" - "to health" is the most common short toast. I don't like it though, since I find it very funny that drinking alcohol is not actually good for health. And the toast is not meant to be funny or sarcastic, therefore it fails in my eyes.
Besides, if Russians are making a toast they usually tell a story that ends in some sort of moral/excuse for merriment and drinking.
Another common toast that I like hearing is "Выпьем за то, что мы здесь собрались, и чтобы чаще собирались!" - "Let's drink to our having gathered here, and to gathering more often!".
The toast in English prononciation - "Vipjem za to, shto mi zdes sobralis, i shtobi chashe sobiralis!"