  We present a novel approach, $\textit{Metric pErTuRbations wIth speCtral
methodS}$ (METRICS), to calculate the gravitational metric perturbations and
the quasinormal-mode frequencies of rotating black holes of any spin without
decoupling the linearized field equations. We demonstrate the method by
applying it to perturbations of Kerr black holes of any spin, simultaneously
solving all ten linearized Einstein equations in the Regge-Wheeler gauge
through purely algebraic methods and computing the fundamental (co-rotating)
quadrupole mode frequency at various spins. We moreover show that the METRICS
approach is accurate and precise, yielding (i) quasinormal mode frequencies
that agree with Leaver's, continuous-fraction solution with a relative
fractional error smaller than $10^{-5}$ for all dimensionless spins below up to
0.95, and (ii) metric perturbations that lead to Teukolsky functions that also
agree with Leaver's solution with mismatches below $1\%$ for all spins below
0.9. By not requiring the decoupling or the angular separation of the
linearized field equations, the METRICS approach has the potential to be
straightforwardly adapted for the computation of the quasinormal-mode
frequencies of rotating black holes of any spin beyond general relativity or in
the presence of matter.
