We extend the common theme of random Apollonian packing of circles to consider orientable grains with a noncircular shape. Systems of up to 10(6) grains are examined for a range of polygonal and elliptical shapes using both the random Apollonian packing model and the new rotational random Apollonian packing model which takes into account the extra rotational degree of freedom of noncircular grains. We identify the constraining length D_c that limits growth of the grain during the packing process and find that a universal relation exists between grain shape and the scaling properties of the system.