I finally looked it up in Websters and noticed the definition/pronunciation key included both accents because the word was originally was pronounced with two long "a" sounds for the accented "e." I think the confusion for us is that in current usage, people pronounce the word incorrectly, using "rez" as the first syllable instead of "ray". It is really fun to say with a French accent. 'Here is my ray-zu-may!'

I agree with Savvy and porsche about the accent changing the vowel sound and pronunciation. I would use Resumé based on my past experience with reading French. Two accents looks like you don't know what you're doing and would be pronounced with the same sound for both "e" characters, and I think most people here would agree that's incorrect. Regarding Brian W.'s post, the sound of the vowels in a word is changed by where you syllabicate a word, which is why we pronounce the "e" sound when a syllable break is added before the final "e" (three syllables total). When used as resume (as in restarting something), there are only two syllables, not three, and in that case, the "e" is silent and is used to make the previous vowel sound long.