A calculational method for Boltzmann's one-velocity, isotropic scattering transport equation is developed for cylindrical rods. The starting point is Peierls' integral equation, and the technique may be interpreted as a moments method or a variational method. Numerical results in the form of graphs are given for a set of standard problems. These problems include volume sources, surface sources, and the critical rod problem. For arbitrary, axially symmetric sources inside or outside the rod, a knowledge of the uncollided flux is sufficient to determine the escape probability from the rod in terms of these standard problems.