What is an appropriate greeting for Yom Kippur? "Hag Sameach" doesn't seem to work. Is it "Have a good fast," or something similar?
I'm interested in knowing Hebrew and English.
Well, Yom Kippur is over for 5776. But the most common greeting is G'mar Chatimah Tova, meaning literally "A good finish of the sealing."
The concept is similar to what is said as part of the Unetaneh Tokef poem, which is one of the highlights of the High Holiday Musaph services. In it, it says, "On Rosh Hashanna it is written, and on Yom Kippur, it is sealed". I.e. on Rosh Hashannah G-d opens the Books of Life and Death. On Rosh Hashannah, mankind's fate is written in one of these books, and on Yom Kippur, the fate is sealed closed.
In addition to "g'mar chasima tova" mentioned by DanF, I've often heard "good yom tov". I often hear them in combination, actually: "good yom tov; g'mar chasima tova".
 It is worth noting that Chabad-Lubavitch folks wish "a chasima and a g'mar chasima tova" — I think until n'ila.