The process of production of three Higgs bosons Ahh in arbitrarily polarized electron-positron annihilation e^- e^+→Ahh is studied within the framework of the Minimal Supersymmetric Standard Model. An analytical expression for the differential effective cross section of the process is obtained, and the left-right A_LR and transverse spin asymmetries A_φ due to the longitudinal and transverse polarizations of the electron-positron pair are determined. The behavior of the spin asymmetries and the differential effective cross section as functions of the emission angles and particle energies is studied in detail. It is revealed that the left-right spin asymmetry A_LR depends only on the Weinberg parameter x_W=sin^2⁡〖θ_W 〗, while the transverse spin asymmetry A_φ is a function of the emission angles and particle energies. The possibility of experimental measurement of the three-boson interaction constants λ_hhh, λ_Hhh and λ_hAA is discussed.