  We present preliminary results of searches for charged B mesons decaying into
the charmless three-body final states hhh, where h = pi or K, using 51.5 fb^-1
of data collected at the Upsilon(4S) resonance with the BABAR detector at the
SLAC PEP-II asymmetric B Factory. No assumptions are made about intermediate
resonances. We measure the branching fractions B(B+ -> K+ pi- pi+) = (59.2 +-
4.7 +- 4.9) x 10^-6 and B(B+ -> K+ K- K+) = (34.7 +- 2.0 +- 1.8) x 10^-6, where
the first error is statistical and the second error is systematic. In the same
study, we do not observe significant signals for the final states B+ -> pi+ pi-
pi+ and B+ -> K+ K- pi+, and therefore provide the 90% confidence upper limits
B(B+ -> pi+ pi- pi+) < 15 x 10^-6 and B(B+ -> K+ K- pi+) < 7 x 10^-6.
