Qudos on the Quenya, PapSett! 
Are you up for Sindarin next?
This is fun.
Ancient Greek:
Εχεις καλους οφθαλμους.
Ekheis kalous ophthalmous.
Modern Greek: someone help me out here…
Εχεις ομορφα ματια.
Ekhis omorfa matia.
Tatar:
Matur küzläreng bar. (familiar)
Matur küzlärengez bar. (polite)
Afrikaans:
Jy het pragtige oë.
Arabic:
‘Indaki ‘aynayn jamilatayn. (said to female)
‘Indaka ‘aynayn jamilatayn. (said to male)
Cantonese:
Ley-ge ngaan ho leng.
Chinese:
您有美丽的眼睛 .
Nin2 you3 mei3li4 de yan3 jing1.
Czech:
Tvoje oči je velma krásna.
Danish:
Du har smukke øjne.
Dutch:
Je hebt mooie ogen.
Esperanto:
Viaj okuloj belas.
Vi havas belajn okulojn.
Estonian:
Sul on ilusad silmad.
Finnish:
Sinulla on kauniit silmät.
French:
Tu as de beaux yeux.
German:
Du hast schöne Augen.
Greek (ancient):
Εχεις καλους οφθαλμους.
Ekheis kalous ophthalmous.
Greek (modern):
Εχεις ομορφα ματια.
Ekhis omorfa matia.
Hebrew:
יש לך עיניים מדהימות
Yesh lakh eynayim madhimot. (said to female)
Yesh lekha eynayim madhimot. (said to male)
Hindi:
Tumhari aankein bohoth sundar hain.
Hungarian:
Gyönyörű a szemed.
Neked gyönyörű szemeid van.
Irish:
Áille is do shúile ort.
Italian:
Hai gli occhi belli.
Japanese:
Kirei na me ga arimasu ne.
Kazakh and Kyrgyz:
Suluu közdering bar.
Klingon:
SoH ghaj ‘IH mInDu’.
Korean:
Noon-ee cham yae-bbeu-neh-yo.
Latin:
Oculos pulchros habes.
Lithuanian:
Tu turi gražias akis.
Malay and Indonesian:
Saudari ada mata-mata cantik. (said to female)
Saudara ada mata-mata cantik. (said to male)
Mongolian:
Chi saikhan nüdüüdtei bain.
Persian:
Chashm ha ye to ziba and.
Pig Latin:
Ou-yay ave-hay eautiful-bay eyes-way.
Pilipino:
Ang ganda ng iyong mga mata.
Polish:
Masz piękne oczy.
Quenya:
Lle hera vanya hendu.
Russian:
У тебя красивые глаза.
U tebya krasivie glaza.
Spanish
Tienes ojos lindos.
Tus ojos son hermosos.
Swahili:
Una macho mazuri.
Swedish:
Du har vackra ögon.
Tamil (literary):
Unakku alakana kankal irukkiratu.
Tamil (spoken):
Unakku azhagana kannugal irukku.
Tatar:
Matur küzläreng bar.
Telugu:
Mee kallu anadamga.
Turkish:
Güzel gözlerin var.
Urdu:
Tumhari ankhen khubsurat hain.
Uyghur:
Gözäl közliring bar.
Uzbek:
Go’zal ko’zlaring bor.
Welsh:
Mae gen ti lygaid hyfryd.
Yucatec Maya:
Hudzutztzcoob a ichoob.
Which opens the age-old Pig Latin debate: should one pig the entire word, or each syllable? Polysyllabic words like “beautiful” are all-too recognisable if obfuscation is the reason for the translation. I propose the alternative “you-bay it-ay ul-fay”. Thoughts?
Fine by me if fine by you! 
I got hands on one more:
Indonesian:
Mata mu sangat indah.
Also, did you know that in Indonesia, you can be called “Superman”. They regard it as kind of a loser name though.
Another example of “Your eyes are very beautiful” wording. So they use the singular even when they mean “eyes.” I think they don’t need to use a plural form as long as plurality is implied somehow. So my example of Malay could be simplified: Saudari ada mata cantik. The mu in your example is a possessive suffix of the second person pronoun kamu (they have way too many words for “you” in that language). So to say “you have beautiful eyes” using the above vocabulary would be Kamu ada mata indah. I know I got the grammar right–but whether anyone actually talks that way is another matter. I think the variations come from how people normally talk, what someone would say in a similar situation, instead of trying for literal translations of the English sentence.
That answers the stated purpose of the OP, which is to actually talk to women. As an aside, though, I’m probably not the only introvert of the SDMB who is more comfortable working her way through all the declensions of a noun and the conjugations of the verb that goes with it, and how they have to fit together, than actually talking to people.
I’m enjoying all this talk of beautiful eyes! I’m not thinking of it as a pickup line. I just like the concept of beautiful eyes.
ERRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
Dork alert!
Run Away!!!
(I snipped parts of the quote, apologies)
It would sound silly to say "Tumhare pas jo aankein hai, botho sundar hai. It means
“The eyes you have, they are very beautiful.”
Which leads to the question - do you have spares?
You can’t really say “You have beautiful eyes”, you have to say “Your eyes are beautiful.”
You can also say “Teri aankein badi neeli hain”
Which means literally You (familiar) have blue eyes",
Very few people actually have blue eyes in most of India, but light-colored eyes are considered more beautiful.
So this colloquially means
“Your eyes are very beautiful”.
In Punjabi, a dialect of Hindi:
“Thwadi ankahn bothan soni hai”
Ehm. What’s with the numbers?
The numbers are Chinese tone indicators.
In Mandarin:
As for Cantonese tones, I was hoping someone could explain clearly. I tried to get a grip on Cantonese tones but was baffled. The books I read lacked audio. I think if any language learner needed audio for anything, it would be Cantonese tones. And Hmong. Do not attempt Hmong without audio.
I think in Sindarin it would be
Gerig hin bein.
Again, I think. I’m not sure I pluralized ‘bein’ correctly.
Unusually, the Quenya sounds so much prettier. I’d translate it into Adunaic as I like that language more – it isn’t very pretty but it sounds neat – but my rather rudimentary Adunaic dictionary doesn’t have ‘eye’ in it.
Don’t cry. A literal translation of “Your eyes are beautiful” would be “Gyönyörű a szemed.” But like you I would have gone for “Nagyon szép a szemed” as the more conversational. My wife says it to me all the time (Debreceni lany, born and raised).
Bun kulia soma tukus.
Awesome! Just started seeing a beautiful Polish woman… sure I can work both of these into conversation… yes, this one thread right here just made the annual SDMB fee more tha worth it… 
Yeah I second that. Good work people.
Good luck with the “dupe” one Crandolf.
Dear Sir or Madam,
I have an urgent need to inform you of my desire to tell you in the strictest of confidence that, by the grace of God and his son Jesus Christ, your esteemed presence has the sum of TWO (2) eyes of extraordinary beauty.
Thank you, and God Bless as I wait for your humble response.
(Nigerian)
Binary;
0101100101101111011101010010000001101000011000010111011001100101001000000110001001100101011000010111010101110100011010010110011001110101011011000010000001100101011110010110010101110011
c’mon, someone had to do it
Thanks… I’ve already said the same essentialy in English so this isn’t going to be shocking or anything…
Ok, i did butcher it.
Nee kallu andamga unayee (familiar)
Mee kallu andamga unayee (formal)
Well, that one version in colloquial speech in a butchered romanization. But don’t take my word for it.
I opened this thread to add the two languages I know: Italian and Estonian. I’m not surprised that Italian was already there, but I was rather surprised to see Estonian.
I don’t speak either fluently, so I can’t comment on how natural the phrases given here sound, but I think they are pretty good.
I should note that the formal Italian would be ha gli occhi belli and IIRC in the south you might say avete gli occhi belli for the informal.
In Estonian the formal equivalent of sul on ilusad ilmad would be teil on ilusad ilmad. I think you might also be able to say sul/teil on kaunid silmad similar to Johanna’s Finnish example. Another possibility is sul/teil on kenad silmad. Kenad is, I believe, best translated as ‘pretty’ or ‘cute’.
One thing to bear in mind is that Estonian doesn’t voice stops, so the sounds written ‘b’, ‘d’ and ‘g’ are pronounced ‘p’, ‘t’ and ‘k’ respectively and that stress always falls on the first syllable. So, sul on kaunid slimad, e.g., is actually pronounced ‘sul on kaunit silmat’.