The effects of temperature-induced electronic Stoner excitations on the spin-wave spectrum of one- and two-dimensional 3d transition metals are investigated in the framework of ab initio density-functional theory by using a generalized-gradient approximation to the exchange and correlation energy. The dispersion relation of frozen-magnon spiral states is calculated as a function of spin-wave vector →q and electronic temperature Te for Fe, V, Co, and Ni monoatomic chains and for periodic Ni square lattices. The resulting temperature dependence of the magnetic order and its stability are analyzed in some detail. A variety of element-specific behaviors are identified including ferromagnetic, antiferromagnetic, and spiral orders. In all considered cases, the local magnetic moments are found to be remarkably stable both as a function of →q and Te. Effective exchange interactions Jij between these moments are derived in the framework of a classical Heisenberg model by fitting the ab initio electronic free energy Ω as a function of →q. The changes in the magnetic order as a function of Te are interpreted as the result of the interplay between competing Jij among different nearest neighbors. Finally, the electronic internal-energy and entropy contributions to the magnon dispersion relation Ω(→q) are discussed.