We have analyzed the expressed manifestation of the anisotropy of surface energy density in the dynamics of ultrathin nanowires, which break up into disjointed clusters when annealed below their melting temperature. The breakup process is studied for different temperatures and orientations of the nanowire axis relative to its internal crystal structure using the Monte Carlo kinetic method. We have also presented an approximate analytical model of the instability of nanowires. Generally, the interpretation of experimental results refers to the theoretical model developed by Nichols and Mullins, which is based on conceptions about the Rayleigh instability of liquid jets. In both cases, the theories -which do not take into account the anisotropy of surface energy density -predict the breakup of a nanowire/liquid jet with radius into fragments with an average length Λ = 9 . However, the observed value, Λ/ , often deviates from 9 either to lower values or to substantially greater ones (up to 24-30). Our results explain various observed features of the breakup and the significant variations in the values of its parameter Λ/ depending on experimental conditions. In particular, the ambiguous role of exchange by atoms of the surface of a nanowire with the surrounding layer of free atoms formed as a result of their rather intensive sublimation, which occurs in a number of cases, has been investigated. We have shown that this exchange can lead both to a decrease, and to a significant increase, in the parameter Λ/ . The obtained results could be potentially useful in applications such as the development of optical waveguides based on ordered nanoparticles chains.