  We investigate the gravitational radiation emitted by an isolated system for
gravity theories with Lagrange density $f(R) = R + aR^2$. As a formal result we
obtain leading order corrections to the quadrupole formula in General
Relativity. We make use of the analogy of $f(R)$ theories with scalar--tensor
theories, which in contrast to General Relativity feature an additional scalar
degree of freedom. Unlike General Relativity, where the leading order
gravitational radiation is produced by quadrupole moments, the additional
degree of freedom predicts gravitational radiation of all multipoles, in
particular monopoles and dipoles, as this is the case for the most alternative
gravity theories known today. An application to a hypothetical binary pulsar
moving in a circular orbit yields the rough limit $a \lesssim
1.7\cdot10^{17}\,\mathrm{m}^2$ by constraining the dipole power to account at
most for 1% of the quadrupole power as predicted by General Relativity.
