Afrikaans
haas (hare) or konyn

Albanian
Lepur
Lepurush (for little one, or for a bunny)

Arabic
arneb
araanib

Armenian 
Nabastak

Bangali 
chorgosh

Bahasa Malaysia
arnab

Basque 
konejoak

Bavarian
kinihås

Bulgarian
zayek (rabbit, hare)

Burmese
Youn 

Catalan
conill 

Chinook 
Quetshadee

Czech
kralik (long "a" and "i") 
kralicek (little rabbit) It's also the word for 'young king'

Cantonese
pak toi (white rabbit) 
yah toi (wild rabbit) 
toi bao bao (baby rabbit)

Cheyenne
vohkoohe

Chinese (Madarin) 
tuzi: rabbit 
baitu: white rabbit 
xiao baitu: little white bunny 

Columbian
bunnylingus Rabbit is regarded as a sacred creature in Columbia.

Croatian
kunic 
zets

Danish
kanin (pronounced khaneen)

Dutch 
konijn or konijntje (Bunny)
"nijntje" (say: naintche),is used when you really love your bunny. 
"nijntje" is also THE most famous cartoon rabbit in Holland, very popular with kids and made by Dick Bruna.

English (Modern)
rabbit
hare
bunny
bunny rabbit
jack rabbit

English (Old) 
coney

Eskimo 
ukalerk

Esperanto
kuniklo

Estonian
kodu-janes ("tame hare")

Finnish
jänis
"kani" is the tame rabbit that doesn't survive in Finland outdoors during winter as it does in England. Then we have also "rusakko" that is larger than "jänis" and stays greyish-brown through the winter ("jänis" turns to white with black eartips). And last but not least we have "pupu", Finnish equivalent to bunny. "vemmelsääri" and "ristihuuli" are more "poetic" names for rabbits. 

Flemish
konijn 
keun

 French 
le lapin or la lapine a tame rabbit is lapin de clapier a young rabbit is le lapereau

German 
das kaninchen

Greek (Modern) 
kouneli

Greek (Classical) 
lagos

Hawaiian
lapaki

Hebrew
Arnevet 
arnavon/arnavoni (little sweet bunny)

Hindi 
khargosh

Hungarian
nyúl or házinyúl 
bunny: nyuszi (no accent)
Note: nyúl is also a verb meaning 'to reach', so "Nyúlok a nyúlért" means "I reach for the rabbit"!

Icelandic 
kanina

Indian (tamil) 
muyal (pronuonced "moo yell")

Indonesian
kelinci

Irish 
coinin

Italian
il coniglio 
bunny is il coniglietto

Japanese
usagi 

Kazakhstan Republic (former USSR ) 
Kenek

Kernewek/Cornish
conyn, conynas (plural)

Kenyan
In Kiswahili (the national language of Kenya): sungura
In Luo (the tribal language of people around Lake Victoria): apuoyo

Korean
toki (pronounced Toe Key) There's a popular Korean children's song that everyone learns when they're little called San Toki (Mountain Rabbit). Translated it goes something like this:
Mountain Rabbit, Rabbit, where are you going? As you hop hop hop, where are you going?

Latin 
cuniculus ('lepus' for hare). It was also used for soldiers who dug tunnels. There is a wonderful word in Latin, 'cuniculosus', which means 'full of rabbits'.

Latvian
trusis 
zakis (hare)

Lithuanian
kralikas (pronouced "kraa-lick-us" like "bus") 
a small bunny rabbit: zuikutis (pronouced "zook-utt-tiss") 

Lwoo
apwoyo

Malagasy 
bitro

Malay 
kelintji

Maltese
fenek

Manx Gaelic
conneeyn or 
conning (bunny)

Maori (New Zealand) 
raapeti ("i" said like long "e" and "r" is rolled)