We build an experiment of optical tweezers based on the use of an inverted optical microscope for manipulating microsized single crystals, which are made of an organic dye and parallelepiped in shape. The microcrystals are directed so that their long axis is in the axial direction of the trapping beam. Their short axis follows the direction of the linear polarization of the beam. In circular or elliptic polarization, the crystals are spontaneously put in rotation with a high speed of up to 500 turns per second. It is the first time, to the best of our knowledge, that such a result is reported for particles of the size of our crystals. Another surprising result is that the rotation speed was first increased as expected by increasing the incident power, but after passing by a maximum it decreased until the complete stop of rotation, whereas the power continued growing. This evolution was not reversible. Several hypotheses are discussed to explain such behavior.