Doublet bands observed in 124,126,130,132 Cs isotopes are studied using the recently developed multi-quasiparticle microscopic triaxial projected shell model (TPSM) approach. It is shown that TPSM results for energies and transition probabilities are in good agreement with known energies and the recently measured extensive data on transition probabilities for the bands in 126 Cs. In particular, it is demonstrated that characteristics transition probabilities expected for the doublet bands to originate from the chiral symmetry breaking are well reproduced in the present work. The calculated energies for 124,130,132 Cs are also shown to be in reasonable agreement with the available experimental data. Furthermore, a complete set of the calculated transition probabilities is provided for the doublet bands in 124,130,132 Cs isotopes.