Magnon drag is shown to dominate the thermopower of elemental Fe from 2 to 80 K and of elemental Co from 150 to 600 K; it is also shown to contribute to the thermopower of elemental Ni from 50 to 500 K. Two theoretical models are presented for magnon-drag thermopower. One is a hydrodynamic theory based purely on nonrelativistic, Galilean, spin-preserving electron-magnon scattering. The second is based on spin-motive forces, where the thermopower results from the electric current pumped by the dynamic magnetization associated with a magnon heat flux. In spite of their very different microscopic origins, the two give similar predictions for pure metals at low temperature, allowing us to semiquantitatively explain the observed thermopower of elemental Fe and Co without adjustable parameters. We also find that magnon drag may contribute to the thermopower of Ni. A spin-mixing model is presented that describes the magnon-drag contribution to the anomalous Nernst effect in Fe, again enabling a semiquantitative match to the experimental data without fitting parameters. Our paper suggests that particle nonconserving processes may play an important role in other types of drag phenomena and also gives a predicative theory for improving metals as thermoelectric materials.