Lifetimes of excited states are one of the most powerful observables for nuclear structure. Lifetimes directly relate to the transition probability between states, which in turn provides information on nuclear collectivity, deformation, band structure and single particle states. These are exactly the type of information needed in the hunt for triaxiality in the nuclear landscape. This long-proposed axially asymmetric deformation has recently been the focus of many experimental and theoretical efforts, and while a handful of triaxial nuclei have been observed, rigid triaxial deformation in the nuclear ground state has not been established. Far from stability, nuclear structure and the onset of deformation evolves rapidly. Reliable structure information is crucial in understanding this shape evolution across isotopic and isobaric chains, and triaxiality has been proposed as the shape-transition mechanism responsible for the behavior seen in rapid transition regions. In particular the A ~ 110 nuclei exhibit a brisk onset of deformation. Signatures of shape-phase transitions, triaxial deformations, shape coexistence and oblate configurations have been reported within a small window of the nuclear landscape, from both theory studies and experimental measurements. In this thesis, we investigated the behavior of the A=109 beta decay chain, specifically the ruthenium and palladium isotopes, from the fission of 238U at the Univ. of Jyvaskyla IGISOL facility. Level lifetimes and gamma-ray transitions were measured with a multi-detector array consisting of of 2 HPGe detectors, 2 LaBr scintillators and 1 plastic scintillator. Triple coincidence $\ eta$-$\gamma$-$\gamma$ events were recorded and used to construct/check level schemes, and extract picosecond-range level lifetimes via the fast-timing method pioneered by H. Mach. Our results include new data for the low energy level schemes of 109Tc, 109Ru and 109Pd, including the first lifetime measurements in 109Tc, where three excited state half-lives were measured; 3 new levels, 31 new transitions and 9 new half-lives in 109Ru; and 1 new level, 7 new transitions and 8 new lifetimes in 109Pd. These results are discussed in the context of the Ru and Pd isotopic chains. Calculations with the Core-Quasiparticle Coupling Model for both the case of rigid triaxial rotational deformation and $\gamma$-soft collective deformation are used to compare the level-energies, angular momentum + parity and transition rates for 109Ru and 109Pd. This comparison provides evidence of rigid triaxiality in the A = 109 isotopes.