  We obtain dimension free estimates for noncommutative Riesz transforms
associated to conditionally negative length functions on group von Neumann
algebras. This includes Poisson semigroups, beyond Bakry's results in the
commutative setting. Our proof is inspired by Pisier's method and a new
Khintchine inequality for crossed products. New estimates include Riesz
transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's
conjecture fails) or to the word length of free groups. Lust-Piquard's work for
discrete laplacians on LCA groups is also generalized in several ways. In the
context of Fourier multipliers, we will prove that H\"ormander-Mihlin
multipliers are Littlewood-Paley averages of our Riesz transforms. This is
highly surprising in the Euclidean and (most notably) noncommutative settings.
As application we provide new Sobolev/Besov type smoothness conditions. The
Sobolev type condition we give refines the classical one and yields dimension
free constants. Our results hold for arbitrary unimodular groups.
