The first words that occurred to me were names: Clytemnestra and Odysseus, from Greek myth. That turned out to be a good hint, because almost all of the other words I could think of also derived from Greek. This is a little ironic because, as far as I know, the Greek character for y, upsilon, is pronounced more like "u" than our long "i" sound. I did manage to find 6 English words with y in the middle, but I'll save those for the end of this post (see how many you can get -- I'm sure I missed plenty of others).
Most of the Greek-derived words used prefixes that contained the letter y. Some obvious ones include "hyper-" (above), "hypo-" (below), "sym-" or "sy-" (together), "psycho-" (soul), "physi-" (nature), "cycl-" (circle), and "hydro-" (water). Here are some less common ones that you might see:
|myo-||muscle||myocardial infarction (heard attack)|
"Thyme," which looks Anglo-Saxon, is actually derived from Greek. But "rhyme," which looks Greek, is genuinely English. It was originally spelled "rime," but apparently adopted its modern spelling in imitation of the Greek word "rhythm."
Another y word not derived from Greek is wyvern, a sort of dragon. The other English words containing y that I was able to come up with all follow the same pattern: dye, lye, rye, and eye.
In addition to my own brainstorming, I found a wonderful site that actually allows one to search on words containing y. I'm sure that anyone who searches that list is bound to discover more y words without Greek origin. Several that I see come to English through Spanish: arroyo, canyon, coyote. Another foreign word that we have come to hear a lot in the past 30 years is ayatollah. Somehow, it doesn't bother me to have missed those words, but I am aghast that I didn't think of crayon.