We study inclusive di-pion decays using a sample of 108 x 10^6 Upsilon(3S) events recorded with the BABAR detector. We search for the decay mode Upsilon(3S) --> pi pi- h_b(1P) and find no evidence for the bottomonium spin-singlet state h_b(1P) in the invariant mass distribution recoiling against the pi pi- system. Assuming the h_b(1P) mass to be 9.900 GeV/c^2, we measure the upper limit on the branching fraction B[Upsilon(3S) --> pi pi- h_b(1P)] < 1.2 x 10^{-4}, at 90% confidence level. We also investigate the chi_{bJ}(2P) --> pi pi- chi_{bJ}(1P), Upsilon(3S) --> pi pi- Upsilon(2S), and Upsilon(2S) --> pi pi- Upsilon(1S) di-pion transitions and present an improved measurement of the branching fraction of the Upsilon(3S) --> pi pi- Upsilon(2S) decay and of the Upsilon(3S)-Upsilon(2S) mass difference. Show Partial Abstract