This was something my Greek professor (years ago) mentioned in passing. While the ancient Greek I learned, and most people learned, has pitched-based stresses, an even more archaic form used vowel lengths in place of stresses as we know them. Vowel lengths are just what they sound like - vowels pronounced for a little bit longer a period of time. Later Greek has some remnants of them such as the letters Omega ('big O'

and Omicron ('small O'

.
As far as the pronunciation of 'Padmé' goes, first of all consider the letter 'e' as pronounced in French or Spanish: 'eh.' Now draw that sound out for a longer period of time: 'ehhhhhh...' You should sound a little bit like the Fonz; it's a very minor change to get 'ay'. (There are ways of distinguishing the differences between those sounds, but let's leave it at that for the moment.) Thus, while Padme (no accent) would be pronounced 'pad-meh', Padmé is 'pad-may'.
(If any better-informed linguists want to respond to this, please be kind. The various language and linguistics courses I took were all more than a decade ago.)