Yes; the following construction does this with 12 congruent pieces. Suppose the disk has radius r. Divide the perimeter of the disk into 6 equal arcs and draw circular arcs of radius r from the disk's center to each of these division points, with each of these arcs concave in the clockwise direction. This divides the disk into 6 congruent pieces which are point symmetric about the disk's center. Note that these pieces each have an axis of symmetry about which the disk's center and one of the division points on the perimeter are symmetric. For 5 of the pieces, cut the piece in half with a circular arc of radius r from disk's center to the midpoint of the perimeter arc of that piece (again concave in the clockwise direction). Cut the 6th piece in half with the reflected image of the arc from the disk's center to the midpoint of the perimeter arc, reflected across the axis of symmetry of that piece. This final cut then is a circular arc from a division point on the perimeter of the disk to the midpoint of one of the original cuts.

--Travis Fisher (fisher@math.psu.edu)