Lifetimes of the ${2}_{1}^{+}, {4}_{1}^{+}, {6}_{1}^{+}, {2}_{\ensuremath{\gamma}}^{+}$, and ${3}_{\ensuremath{\gamma}}^{+}$ states in $^{104,106}\mathrm{Ru}$ were measured by using the recoil-distance Doppler-shift technique and the Cologne Plunger device. Low-lying excited states in both nuclei were populated in a $^{104}\mathrm{Ru}(^{18}\mathrm{O},^{18}\mathrm{O})^{104}\mathrm{Ru}$* inelastic scattering and in a $^{104}\mathrm{Ru}(^{18}\mathrm{O},^{16}\mathrm{O})^{106}\mathrm{Ru}$ two-neutron transfer reaction using the Cologne FN Tandem accelerator. The experimental energy levels and deduced electromagnetic transition probabilities are compared in the context of $\ensuremath{\gamma}$ softness and the mapped interacting boson model with input from the microscopic self-consistent mean-field calculation using a Gogny interaction. The newly obtained results for the $\ensuremath{\gamma}$ band give a more detailed insight about the triaxial behavior of $^{104,106}\mathrm{Ru}$. The results will be discussed in the context of $\ensuremath{\gamma}$ soft and rigid triaxial behavior which is present in the neutron-rich Ru isotopes.