There are two types of divination that I think work well with RPGs. One is what I'll call the "Magic 8-Ball" technique: the method of forecasting the future provides only general answers to yes/no questions, which is often the case with actual divination techniques; the answer is as often as not a "maybe," giving the referee flexibility to opt out if a question is truly in doubt. ("Will we get rich?" "Outlook not so good.")
A second type is to think of the Delphic oracle. Prophesies given at Delphi were so elaborate and open-ended in their phrasing that there were always several potential interpretations. Many famous myths such as Oedipus begin with oracular utterances where attempts to avoid the disastrous consequences actually cause them to come true; the myth of Oedipus is a famous example of this.
Beyond this, we should ask: what is fortune-telling actually useful for in RPGs? In a dungeon crawl or sandbox, it may seem odd or even superfluous to actually predict the future. But I think the two types of fortune telling have several good roles in an RPG context.
First, there is the use of fortune-telling to drop clues and ideas about what is in a dungeon or sandbox type of environment. For instance, a character consulting a fortune teller might get a cryptic hint that "the green-eyed gems are jealous and plot destruction" if there is a trap involving emeralds at some level of a dungeon; knowing the nature of players there will be some inevitable back-and-forth before they actually settle on what to do about the green-eyed gems of destruction, perhaps looking for a statue with green eyes or some such.
Second, there is the instant adventure seed. This is probably richer for sandbox play, but a hint in divination about what lies beyond the Black Mountains is bound to drive curiosity wild. It's one of the relatively few places where it doesn't seem out of place to just stick an arrow on the map and say "Go there!" - since after all, the characters are literally asking for it. Combining this with the Magic 8-Ball method would also do a fine job of livening up a game that has gotten a bit flat and stale from the GM's side - using the question-and-answer format as a place to make some decisions on your feet, as it were.
Third, there are the prophesies that deal with the "big" things in the game. If a megadungeon has an ultimate goal, like finding Zagyg Yragerne in Castle Greyhawk, then a good cryptic prophecy should be just the thing for it. In general such divination should be the most general type, since after all the actual outcome of the final encounter should never be a certain thing. This is one way to give a campaign some shape and depth beyond "The Quest for More Money."
So I do see a strong role for divination magic in D&D and I think it'll be interesting to get a bit more into the systems for it in Demon-Haunted Lands. I'd like to hear more about how folks have used fortune telling in RPGs before - both good and bad, if there are horror stories that can serve as cautionary tales.