A theory of singular integrals with monogenic kernels on star-shaped Lipschitz surfaces inR ⁿ is established. The class of singular integrals forms an operator algebra identical to the class of bounded holomorphic Fourier multipliers, as well as to the Cauchy-Dunford bounded holomorphic functional calculus of the spherical Dirac operator. The study proposes a new method inducing Clifford holomorphic functions from holomorphic functions of one complex variable, by means of which the study on the sphere is reduced to that on the unit circle.