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The magnetocrystalline anisotropy energy (MAE) for ferromagnetic and antiferromagnetic freestanding monowires of transition metals is investigated on the basis of first-principles calculations. Across the series, the easy axis of the magnetization oscillates between two directions: perpendicular and along the wire axis. The largest values of the MAE occur at the end of the series. Giant values of can be obtained upon stretching Ru or Rh wires. Ru and Pd chains change the magnetization direction upon wire stretching, opening new perspectives in controlling the spin-dependent ballistic conductance in these structures.