We report results of a study of charmless B meson decays to three-body Kππ,KKπ and KKK final states. Measurements of branching fractions for B decays to K+(0)π+π−,K+K+K−,K0K+K−,K+K0SK0S and K0SK0SK0S final states are presented. The decays B0→K0K+K−,B+→K+K0SK0S and B0→K0SK0SK0S are observed for the first time. We also report evidence for B+→K+K−π+ decay. For the three-body final states K0K+π−,K0SK0Sπ+,K+K+π− and K−π+π+, 90% confidence level upper limits are reported. Finally, we discuss the possibility of using the three-body B0→K0SK+K− decay for CP violation studies. The results are obtained with a 78 fb−1 data sample collected at the Υ(4S) resonance by the Belle detector operating at the KEKB asymmetric energy e+e− collider.