We shall provide dimension free estimates for Riesz transforms on group von Neumann algebras. Our results include new Riesz transforms beyond Bakry's results. This includes the Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ or to the word length of free groups. Lust-Piquards work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we have found that any H"ormander-Mihlin multiplier arises as a Littlewood-Paley average of our Riesz transforms. As an application, we obtain new Sobolev/Besov type smoothness conditions for group von Neumann algebras. Our results hold for arbitrary unimodular groups. This is joint work with M. Junge and T. Mei.