We present measurements of branching fractions and charge asymmetries for charmless B-meson decays to three-body final states of charged pions and kaons. The analysis uses 81.8 fb-1 of data collected at the Υ(4S) resonance with the BABAR detector at the SLAC PEP-II asymmetric B Factory. We measure the branching fractions B(B+→π+π-π+)=(10.9±3.3±1.6)×10-6, B(B+→K+π-π+)=(59.1±3.8±3.2)×10-6, and B(B+→K+K-K+)=(29.6±2.1±1.6)×10-6 and provide 90% C.L. upper limits for other decays. We observe no charge asymmetries for these modes.
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