The interplay of inertia and elasticity is shown to have a significant impact on the transport of filamentary objects, modeled by bead-spring chains, in a two-dimensional turbulent flow. We show how elastic interactions among inertial beads result in a nontrivial sampling of the flow, ranging from entrapment within vortices to preferential sampling of straining regions. This behavior is quantified as a function of inertia and elasticity and is shown to be very different from free, noninteracting heavy particles, as well as inertialess chains [Picardo et al., Phys. Rev. Lett. 121, 244501 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.244501]. In addition, by considering two limiting cases, of a heavy-headed and a uniformly inertial chain, we illustrate the critical role played by the mass distribution of such extended objects in their turbulent transport.