Let H be a complex, separable Hilbert space (of finite or infinite dimension), and let U(H) denote the group of unitary operators on H. In the finite-dimensional setting, we prove that every unitary operator of determinant one can be expressed as the product of two operators, each unitarily equivalent to the n × n unitary cycle. In the infinite-dimensional setting, we prove that every unitary operator U is a product of three operators, each unitarily equivalent to the bilateral shift, and if the spectrum of U has nonzero Lebesgue measure, then U is a product of two operators, each unitarily equivalent to the bilateral shift. This work is joint with Laurent Marcoux, Matjaž Omladič and Heydar Radjavi.