  The detection of gravitational waves from compact binary mergers by the
LIGO/Virgo collaboration has, for the first time, allowed for tests of
relativistic gravity in the strong, dynamical and nonlinear regime. Outside
Einstein's relativity, spinning black holes may be different from their general
relativistic counterparts, and their merger may then lead to a modified
ringdown. We study the latter and, for the first time, derive a modified
Teukolsky equation, i.e., a set of linear, decoupled differential equations
that describe dynamical perturbations of non-Kerr black holes for the radiative
Newman-Penrose scalars $\Psi_0$ and $\Psi_4$. We first focus on non-Ricci-flat,
Petrov type D black hole backgrounds in modified gravity, and derive the
modified Teukolsky equation through direct decoupling and through a new
approach, proposed by Chandrasekhar, that uses certain gauge conditions. We
then extend this analysis to non-Ricci-flat, Petrov type I black hole
backgrounds in modified gravity, assuming they can be treated as a linear
perturbation of Petrov type D, black hole backgrounds in GR by generalizing
Chandrasekhar's approach, and derive the decoupled modified Teukolsky equation.
Our work lays the foundation to study the gravitational waves emitted in the
ringdown phase of black hole coalescence in modified gravity for black holes of
any spin. Our work can also be extended to compute gravitational waves emitted
by extreme mass-ratio binary inspirals in modified gravity.
