The general three-dimensional periodic orbits around asteroids were investigated to study orbital behaviours in the vicinity of irregular gravitational bodies. The orbital patterns around periodic orbits were determined through decomposition into corresponding local manifolds. In this paper, a topological classification of periodic orbits is presented to discriminate the stability and dynamical behaviours of neighbours. A hierarchical grid searching method was developed for systematically searching three-dimensional periodic orbits around irregular bodies. The method was used to generate 29 basic families around the asteroid 216 Kleopatra as an example. By calculating the characteristic multipliers of periodic orbits, numerical evidence was generated to describe the denseness of periodic orbits around asteroids and to demonstrate the remarkable asymmetry among these orbits. The dependence of the topology of the periodic orbits on the Jacobi integral was examined, revealing the evolutionary features of the stability and orbital patterns nearby. The results can be used to assess the environment around 216 Kleopatra, and this method can be useful in the studies of other asteroids.