An iterative method we previously proposed to compute nuclear strength functions [ Toivanen et al. Phys. Rev. C 81 034312 (2010)] is developed to allow it to accurately calculate properties of individual nuclear states. The approach is based on the quasiparticle random-phase approximation (QRPA) and uses an iterative non-Hermitian Arnoldi diagonalization method where the QRPA matrix does not have to be explicitly calculated and stored. The method gives substantial advantages over conventional QRPA calculations with regards to the computational cost. The method is used to calculate excitation energies and decay rates of the lowest-lying 2+ and 3− states in Pb, Sn, Ni, and Ca isotopes using three different Skyrme interactions and a separable Gaussian pairing force.