The work is concerned with the determination of effective conductivities and field potentials of matrix-based composites consisting of periodic arrays of cylinders which are cylindrically orthotropic and exponentially graded along the radial direction. We generalize Rayleigh's method to account for the periodic arrangements of these cylinders. The potential field and effective conductivities of composite systems were calculated to a very high order to achieve a sufficient accuracy. We find that the cylindrical orthotropy of the inclusions has a dramatic effect on the potential field of the inclusions. In addition, we discuss the effect of the grading factor on the effective conductivity. Interestingly, we find that when the inclusions are purely cylindrically orthotropic, their effects can be fully described by homogeneous isotropic cylinders. This equivalent isotropic conductivity is simply the geometric mean of the radial and tangential conductivities of cylindrically orthotropic cylinders.