I went to the OED, and it directly quotes the original coining of the word, which conveniently enough includes a pronunciation guide.
1976 R. DAWKINS Selfish Gene xi. 206 The new soup is the soup of human culture. We need a name for the new replicator, a noun which conveys the idea of a unit of cultural transmission, or a unit of imitation. "Mimeme" comes from a suitable Greek root, but I want a monosyllable that sounds a bit like "gene". I hope my classicist friends will forgive me if I abbreviate mimeme to meme... It should be pronounced to rhyme with "cream".