Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL$_2$ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL$_3$ have been considered, in which analogous GL$_3$-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL$_3(\mathbb Z)$. We give formulas for the first three moments and a nontrivial bound for the fourth.