As is well known, Ecclesiastical Latin has its own pronunciation--or family of pronunciations, if we include regional variants, the prevailing one being the italianate pronunciation. This pronunciation is notably different from what scholars since Erasmus' De recta latini graecique sermonis pronutiatione (1528) have been telling us would have been the original, historical pre-Christian pronunciation of Latin (of the weni-widi-wiki variety). Of course, we have no certainty about the way Latin was really spoken in classical times, simply because we have no audio recordings of it. And the written clues the ancients left us only go so far. The reconstruction is inevitably somewhat artificial.
Moreover, although few scholars would acknowledge it, the matter is significantly aggravated by the fact that English-speaking scholars are particularly bad at capturing the phonetic nuances of Mediterranean languages. (You can make a Mexican laugh by asking one of these professors to sing "La Cucaracha" as best they can.) Most native English speakers, no matter how smart, have no business trying to reconstruct what Latin would really have sounded like around the First Century AD. The way their brain-tongue neurons connect is dramatically un-Mediterranean.
Most people in the Church are aware of these differences and have sensibly embraced the ecclesiastical pronunciation of Latin as being just the way Latin is realistically pronounced. We speak Latin with the pronunciation we have gratefully received from our ancestors in the Faith. It may not be historically rooted in Cicero, Virgil, or Ovid, but it is certainly what we have been handed down. This is so much the case that, for Catholics who encounter Latin in the liturgy on a daily basis, using a 'classical' Latin pronunciation in Church would be laughable. And of course in academic circles, these same Catholics use the ecclesiastical pronunciation, even when they are reading classical texts. The revival of Latin that we have been witnessing in the Church is happily based on the received traditional pronunciation.
Yet what few Catholics, scholars or otherwise, realize is that Greek, too, is an ecclesiastical language. Yes, there are Greek Catholics, Greek liturgies, with Greek prayers, and even Greek chant. And no, it's not just the Eastern Orthodox who use these things. It is all originally Catholic, and continues to be essentially Catholic.
And just like in Latin, in Greek there is a huge difference between the way it is pronounced in actual practice in the Church and the way Erasmus and his modern followers suppose it was spoken in classical and early Christian times. One of the most notable differences has to do with the issue of iotacism (in ecclesiastical Greek ι, η, υ, ει, οι, ηι, υι are all pronounced |i|, like a iota, so you hear that sound a lot). The phonetic details lie outside of the scope of this post, but the main point is that if you are exclusively familiar with one way of doing it, hearing the other way of doing it can be highly distracting, funny, and even annoying or simply unintelligible.
But if you call the Latin rites your liturgical home, and are blessed to have studied Greek (or at least the rudiments of it), chances are you are unaware of the ecclesiastical pronunciation. You probably learned the "Erasmian" pronunciation; you pronouce Greek with the (purportedly) 'classical', bookish pronunciation of secular academics. Even if you took courses in Biblical Greek, you were taught to use this half-made-up, half-historical, wholly-dead pronunciation. If you are a philosopher, you habitually talk about episteme, not realizing Greek philosophers actually pronounce it "eh-pis-tee-mee." And perhaps most embarrasing of all, you were taught to pronounce Κύριε ελέησον (Kýrie eléison), as "Koo-ri-eh eh-leh-eh-son," despite your liturgical instincts telling you otherwise. Something is clearly wrong.
So if the above paragraph describes you, then I have a challenge for you. Christianize your Greek pronunciation. Make it more Catholic. Make it traditional. Not historically accurate, or authentically ancient (or whatever). But authentically Byzantine, the way it was handed down within the Byzantine Empire and up to our own day. In a word: traditional.
As is the case with the purportedly classical pronunciation of Latin, the 'classical', or rather Erasmian, pronunciation of Greek, is an artificial reconstruction and has no native speakers on the whole Earth who use it. Much of it is still a matter of debate. Modern Greeks speak modern Greek, which is grammatically distinct from Ancient, Koine, and liturgical Greek, and even when they pronounce their liturgical Greek prayers, they use the received pronunciation. Yes, indeed, "there’s an entire country of people who speak Greek and can’t bear to listen to the awful linguistic barbarity known as Erasmian." Just as no Catholic in their right mind would ever bear to hear chanted "Sal-way Re-gi-na" (even though supposedly that is the correct pronunciation, according to most Latin instructors) so a Greek, no matter how scholarly, would consider it an insult to God's ears to pronounce His revealed Word, originally penned in Greek, in this reconstructed, Erasmian invention. The Catholic spirit is to embrace what one has reverently received. So we sing Κύριε ελέησον (Kýrie eléison).
A good way to start Christianizing your Greek is by learning the Our Father, the Πάτερ ἡμῶν (Páter imón). Below is the text, and an audio recording of an entire Byzantine Divine Liturgy (Holy Mass). Skip to 1:00:44 to hear the recitation of the Πάτερ ἡμῶν. Repeat until you've memorized the text, understood the words, and internalized the beauty of the sounds of Ecclesiastical Greek.
Πάτερ ἡμῶν ὁ ἐν τοῖς οὐρανοῖς,
ἁγιασθήτω το ὄνομά σου,
ἐλθέτω ἡ βασιλεία σου,
γενηθήτω το θέλημά σου,
ὡς ἐν οὐρανῷ και ἐπι τῆς γῆς.
Τον ἄρτον ἡμῶν τον ἐπιούσιον
δος ἡμῖν σήμερον·
και ἄφες ἡμῖν τα ὀφειλήματα ἡμῶν,
ὡς και ἡμεῖς ἀφίεμεν τοῖς ὀφειλέταις ἡμῶν·
και μη εἰσενέγκῃς ἡμᾶς εἰς πειρασμόν,
ἀλλα ῥῦσαι ἡμᾶς ἀπο τοῦ πονηροῦ.
Here are other decent examples, both recited and sung:
And if you try and just cannot get yourself to pronounce it that way, it's ok. Just, whatever you do, please do not pronounce it this way:
I hear that and I immediately imagine God the Father wincing from Heaven.