  We study several aspects of higher-order gravities constructed from general
contractions of the Riemann tensor and the metric in arbitrary dimensions.
First, we use the fast-linearization procedure presented in arXiv:1607.06463 to
obtain the equations satisfied by the metric perturbation modes on a maximally
symmetric background in the presence of matter and to classify
$\mathcal{L}($Riemann$)$ theories according to their spectrum. Then, we
linearize all theories up to quartic order in curvature and use this result to
construct quartic versions of Einsteinian cubic gravity (ECG). In addition, we
show that the most general cubic gravity constructed in a dimension-independent
way and which does not propagate the ghost-like spin-2 mode (but can propagate
the scalar) is a linear combination of $f($Lovelock$)$ invariants, plus the ECG
term, plus a New ghost-free gravity term. Next, we construct the generalized
Newton potential and the Post-Newtonian parameter $\gamma$ for general
$\mathcal{L}($Riemann$)$ gravities in arbitrary dimensions, unveiling some
interesting differences with respect to the four-dimensional case. We also
study the emission and propagation of gravitational radiation from sources for
these theories in four dimensions, providing a generalized formula for the
power emitted. Finally, we review Wald's formalism for general
$\mathcal{L}($Riemann$)$ theories and construct new explicit expressions for
the relevant quantities involved. Many examples illustrate our calculations.
