  We study the evolution of scalar cosmological perturbations in the (1+3)-
covariant gauge-invariant formalism for generic $f(R)$ theories of gravity.
Extending previous works, we give a complete set of equations describing the
evolution of matter and curvature fluctuations for a multi-fluid cosmological
medium. We then specialize to a radiation-dust fluid described by barotropic
equations of state and solve the perturbation equations around a background
solution of $R^{n}$ gravity. In particular we study exact solutions for scales
much smaller and much larger than the Hubble radius and show that $n>2/3$ in
order to have a growth rate compatible with the M\'esz\'aros effect.
