For an abstract self-adjoint operator $L$ and a local operator $A$ we study
the boundedness of the Riesz transform $AL^{-\alpha}$
on $L^p$ for some $\alpha >0$.
A very simple proof of the obtained result is based on the
finite speed propagation property for the solution of the
corresponding wave equation. We also discuss the relation between the
Gaussian bounds and the finite speed propagation property.
Using the wave equation methods we obtain
a new natural form of the Gaussian bounds for the heat kernels for
a large class of the generating operators.
We describe a surprisingly elementary proof of the finite speed propagation
property in a more general setting than it is usually considered in the
literature.
As an application of the obtained results we prove boundedness of the
Riesz transform on $L^p$ for all $p\in (1,2]$ for Schr\"odinger
operators with positive potentials and electromagnetic fields.
In another application we
discuss the Gaussian bounds for the Hodge
Laplacian and boundedness of the Riesz transform on $L^p$ of the Laplace-Beltrami
operator on Riemannian manifolds for $p>2$ .