  This paper proves the $L^p$ boundedness of generalized first order Riesz
transforms obtained as conditional expectations of martingale transforms \`a la
Gundy-Varopoulos for quite general diffusions on manifolds and vector bundles.
Several specific examples and applications are presented: Lie groups of compact
type, the Heisenberg group, SU(2), and Riesz transforms on forms and spinors.
