Riesz(Hilbert) transforms may be the most fundamental fourier multipliers on Rn. I will try to explain our recent discovery (with M. Junge and J. Parcet) that they are actually the `essential' ones in the sense that every Hormander-Milhlin multiplier is an average of `Riesz transforms' associated with semigroups of operators. P. A. Meyer's gradient form gives natural Riesz transforms associated with Markov semigroups. D. Bakry has proved a Riesz transform inequality for semigroups which admit Markov dilations with almost continuous path. Adapting G. Pisier's method to Markov semigroups, we prove D. Bakry's Riesz transform inequality on group von Neumann algebras without the continuity assumption. The flexibility of our results then allows us to write a big class of Fourier multipliers as Littlewood-Paley averages of Riesz transforms.