We show that an anisotropic Mihlin-type condition on the symbol guarantees the boundedness of the associated Fourier multiplier operator onL^p(Rⁿ,X)for 1 < p < ¥L^p(\mathbf{R}^n,X){\rm for} 1\!<\!p\!<\!\infty L ^p (R ⁿ ,X) for 1 < p < ∞ and an arbitrary UMD space X. In many cases, this result can be used as a substitute for the Marcinkiewicz–Lizorkin multiplier theorem, which is invalid in general UMD spaces. An application to anisotropic singular integrals is given.