Mycorrhizae: how the heck do you say "zae" in greek?

Mycorrhizae is an English word, not a Greek one. As such, it doesn't have a "Greek" pronunciation, but rather several English pronunciations. It's true that it's derived from Greek, but that's merely a matter of etymology.

The pronunciation in English of the plural suffix -ae that occurs in Latinate or Greek-derived words is actually an area where there is currently some variance.

The traditional way of pronouncing this digraph is as a long e sound /iː/ (think of Caesar and algae). This system of pronouncing Latinate words appears to still be in use among botanists, judging by the essay "How Do You Say That? A Guide to the Perversities of Botanical Latin", by Tom Fischer, which was published in Horticulture magazine in 2000.

You said that this pronunciation sounds "dorky" to you. Luckily for you, pronouncing these words with an "ay" /eɪ̯/ or "eye" /aɪ̯/ sound is also commonly heard nowadays. So you can really choose whatever pronunciation you want, and you'll have people who pronounce it like you do.

In Greek, it would have been spelled -αι "ai" and pronounced -/aj/ or -/aʲ/ (I think there is debate about this subtle distinction, which does not matter for us here: it sounds like English eye).

In classical Latin, it would have been pronounced the same as in Greek, -/aj/ or -/aʲ/. Almost all Greek words we have came to us through Latin.

In the older international pronunciation of Latin, which was based on the Romance languages but used throughout Europe, it is -/eː/ or -/eɪ/ or something similar (like English say).

English, however, went through several vowel shifts, part of which was that vowels pronounced like /e/ came to be pronounced like /i/. So the pronunciation of the English word written as see was once like modern say, but has shifted to its modern pronunciation, /siː/. Spelling changed more slowly than pronunciation, or not at all.

The English also applied this shift to words of Latin and Greek origin, because such words on -ae were often or normally pronounced /e/ (modern English say) before the vowel shifts, as I said above. It was often spelled -e instead of -ae in manuscripts, but also even in print, at least until the humanists got their way of restoring the classical spelling -ae.

That is why the traditional English pronunciation of -ae in Latin or Greek words has been like /i/ for a long time. If you want to be conventional, I would pronounce it this way, although other pronunciations are not necessarily stupid: they may correspond to different points of view.