I don't claim to speak Greek, but then, who does? Except for the Greeks, of course. But, I once did know a number of Greek people and I believe that "yee-ros" is correct except that you should have a little bit of phlegm in the back of your throat when you say the "yee". Just a little, mind you. (In Greek, I believe there is also a "y" sound, as in "Yanni" (which is "John" in Greek) and it should be distinguishable from the "gy" in "gyro". That's my theory, anyway.) Or, at least, that you've got just a little something or other caught in your throat when you say it. Same goes for the "gli" in Italian.
Now, let's try to explain to a non-English speaker why "phlegm" is pronounced the way it is.
Best to all.
Juan