I can only give you the modern Greek pronounciations, as no one really knows how they were originally pronounced... But I'd say these are a fair assumption:
Erato -> Eh-rah-TO
Aoide -> (This one had me boggled. I'm just going to go by the letters that I see and imagine this is how it's spelt in Greek) Ah-EE-thee (thee as in 'I give to thee')
Euterpe -> Ef-TER-pi
Melete -> Meh-LEH-ti
Melpomene -> Mel-po-MEH-ni
Mneme -> MNI-mi (the first m is not silent)
Ourania -> Ou-rah-NI-ah
Polyhymnia -> Po-lee-heem-NI-ah
Terpsichore -> Ter-psi-HO-ri
Thalia -> THA-li-ah