Many sephardim still have a tradition as to the melody of the ta'amei emet. You can buy recordings of the Moroccan tradition from http://www.tht.co.il/default.asp. If you've visited sepharadi synagogues before, you may recognize the melody -- we use it for Kabbalat Shabbat.
As for the question of why they have ta'amim: the books of the Tanach need some sort of Masoretic punctuation so that we can understand the proper grammatical reading of the text. Whether we have a melody for those ta'amim is really a separate issue. It pays great dividends to take some time to read about the functions of the different ta'amim.