Starting on the night of Sivaratri (March 7), many temples dedicated to Lord Shiva will host dance festivals called Natyanjali where dancers perform to the Lord inside temples. The fests are spread across one evening or a few evenings. The most famed of these Natyanjali festivals are held in Chidambaram and in Thanjavur.
Free and open to all.
Read about them at http://www.kutcheribuzz.com/news/dance/3143-natyanjali-dance-fest-2016