The Hilbert transform H is a basic example of a Fourier multiplier. Riesz proved that H is a bounded operator on Lp(T) for all 1 < p < ∞. We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative Lp spaces. The pioneering work in this direction is due to Mei and Ricard who proved Lp-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. In this talk, we introduce a generalised Cotlar identity and a new geometric form of Hilbert transform for groups acting on R-trees. This class of groups includes free groups, amalgamated free products, HNN extensions, totally ordered groups and many others. Joint work with Adrian Gonzalez and Javier Parcet.