The Mevleviye, one of the most well known of the Sufi orders, was founded in 1273 by Rumi's followers after his death, particularly by his successor Hüsamettin Çelebi who decided to build a mausoleum for Mevlâna, and then Mevlâna's son, Sultan Veled Celebi (or Çelebi, Chelebi) (the word "Çelebi" means " fully initiated"). He was an accomplished Sufi mystic with great organizing talents. His personal efforts were continued by his successor Ulu Arif Çelebi.
The Mevlevi, or "The Whirling Dervishes", believe in performing their dhikr in the form of a "dance" and music ceremony called the sema.
The Sema represents a mystical journey of man's spiritual ascent through mind and love to "Perfect." Turning towards the truth, the follower grows through love, deserts his ego, finds the truth and arrives at the "Perfect." He then returns from this spiritual journey as a man who has reached maturity and a greater perfection, so as to love and to be of service to the whole of creation.
For a nice description of what "turning" is all about watch this video.