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 Upsilon(4S) resonance with the BABAR detector at the SLAC PEP-II asymmetric B Factory. We measure the branching fractions B(B+-->pi(+)pi(-)pi(+))=(10.9+/-3.3+/-1.6)x10(-6), B(B+-->K+pi(-)pi(+))=(59.1+/-3.8+/-3.2)x10(-6), and B(B+-->K+K-K+)=(29.6+/-2.1+/-1.6)x10(-6) and provide 90% C.L. upper limits for other decays. We observe no charge asymmetries for these modes.