First I should advise you that the curious tradition of breaking plates has long been outlawed. It is now illegal to break plates at night clubs and tavrernas. Too many people were being injured by flying glass.
Nowadays people buy trays of flowers and have "flower-throwing" fits where they throw the flowers at the singers or at eachother.
Second I must report that Matt is correct in that most of the tavernas in the Plaka are touristy type places. There is, though, one exception. The name of the Taverna is Xynou and it is written up in all the travel guides to Athens such as Frommer's and Fielding's. This place has an excellent traditional Greek kitchen and features 3 fellows who play the guitar and sing old Greek and international favorites. Unfortunately the owner likes to take the weekend off (both saturdays and sundays). This makes things difficult since you arrive on saturday. You could, though, make reservations for you and your group for the evening you get back from Hydra.
Otherwise there are other excellent tavernas I could recommend which are without music but serve excellent food at very reasonable prices, and then I could recommend other excellent tavernas which have the touristy entertainment, ie. dancing tsoliades, etc., but which serve rather mediocre food at relatively high prices.
The choice is yours.
Regards from sunny Athens, Tom Mazarakis
See also Athens Survival Guide Restaurants