An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multi-dimensional orthogonal expansions on product spaces. This is then applied to obtain \(L^p,\) \(1<p<\infty,\) boundedness of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the \(L^p\) norms of these Riesz transforms are both dimension-free and linear in \(\max(p,p/(p-1)).\) The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.