It's definitely not pronounced like "my thai". That's the name of a drink.
The three letters in the thai word "muay" are more or less the equivalents of the english letters "m", "w" and "y". (In Thai language that's three consonants.) This means that logically it would sound pretty similar to "muy (bien)".
On the other hand it seems like the Thai tend to twist that middle consonant a bit from time to time so that it almost sounds like it turns into a short vowel, like in "must" before the y, maybe because of the fact that the word is entirely made up of consonants. I really don't know what is more correct but I seem to remember hearing Thais pronounce it like "muy bien" more often than not.
Of course, as always when it comes to a far eastern languages, this is not the whole secret about how to pronounce it right. You also need the right intonation, or it will probably mean something completely different. But I don't know anything about that part.