The Romans themselves would have used the IE version, same sound as in my or high. Youleeoos Kighzar. I think that the Seize 'er version is pretty well standard in the English-speaking world, not just the US.
There is a rather welcome movement towards using the Greek versions of names rather than the Latin ones where one is dealing with a Greek original ... not sure where it started, but it does make sense; on that reading, Aesop will be written Aisop and pronounced Eyesop instead of Eesop nowadays, though I don't see it catching on in everyday usage any time soon. Caesar of course never had a Greek pronunciation, so he's out of it!