Most of these words have Greek roots, and most arrived in English during the late middle ages or renaissance. Unfortunately most of them are also sufficiently obscure as not to show up in many dictionaries (plus many of them have multiple spellings), so I wasn’t able to track down as much information about them as I would have liked. Still, they’d be great words with which to impress your friends - especially if you actually managed to use any to foretell the future. But please remember that some of these words are definitely not to be tried at home.
ailuromancy - divination from the actions of cats (used especially to predict weather)
myrmomancy - ants
skatheromancy - tracks of beetles
ophiomancy - snakes
nggam - spiders or crabs (this is a word from the Mambila people of Camaroon and Nigeria)
ololygmancy - howling of dogs
alectryomancy - roosters pecking corn
Those are methods that the animals presumably prefer. There are all too many others that require the death of the unfortunate creature, including
alectormancy - in which the rooster is sacrificed
cephalomancy - in which a donkey’s head is boiled
plastromancy - in which a turtle shell is heated to see the pattern of cracks that form
haruspicy - the general term for the study of entrails, especially livers
anthropomancy - human sacrifice
batraquomancy - frogs… but I don’t know how that’s supposed to work and whether or not the frogs survive the process
gyromancy - spinning around inside a circle marked with symbols or letters until you fall down with dizziness on a symbol
retromancy - looking over your shoulder
oneiromancy - dreaming
belomancy - divination from arrows. There are several methods using marked arrows, including seeing which arrow flies farthest and pulling an arrow at random from the quiver
geloscopy - laughter
fal-gush (Persian, also called cledonism) - finding significance in overheard words
chresmomancy - ravings of lunatics
cheiromancy - palmistry, finding the future revealed on the palm of your hand… and if you like that, why not also try
natimancy, also called rumpology - finding the future revealed in, yes, the rump
tasseomancy - tea leaves from the bottom of your cup (from French for cup)
abacomancy - dust, sand, ashes (from Hebrew for dust)
oomancy - egg cracked into boiling water
margaritomancy - pearls
stercomancy - seeds in bird excrement
trochomancy - wheel ruts
tyromancy - cheese
stichomancy - any book
bibliomancy - the Bible (sometimes used for other books, too)
stoicheomancy - the Iliad, Odyssey, or Aeneid
rhapsodomancy - poetry
shufflemancy - iPod playlist
aleuromancy - divination involving flour, but more importantly also involving dough and messages therein, thus making this the word for telling the future from fortune cookies. Too bad fortune cookies almost never provide actual fortunes!
cromniomancy - the sprouting of onions. There are various ways to do this, but generally you write possible alternatives on a selection of onions and see which one sprouts first.
dracomancy - dragons. I’m intrigued. Do you observe the flight of dragons, or do you have to slice them open and examine their livers? Where do you find these dragons in the first place? And really, it hardly seems much advantage, since it’s just as hard to find the dragons as it is to know the future without them.
moromancy - foolishness. I don’t know how you’re supposed to use foolishness to tell the future, but I can’t help thinking this word really sums up the whole fortune-telling thing with a certain ironic neatness.
[Pictures: Roosters pecking corn, woodcut with watercolor by Gerrit Willem Dijsselhof;
Frogs, woodcut with watercolor by Dijsselhof from Kunst en samenleving, 1894 (Images from Nationale bibliotheek van Nederland);
Palmistry chart, woodcut from Les Oevres by Jean Baptiste Belot, 1640 (Image from The Museum of Ridiculously Interesting Things).]