  Limits on gravitational Cherenkov radiation by cosmic rays are obtained and
used to constrain coefficients for Lorentz violation in the gravity sector
associated with operators of even mass dimensions, including
orientation-dependent effects. We use existing data from cosmic-ray telescopes
to obtain conservative two-sided constraints on 80 distinct Lorentz-violating
operators of dimensions four, six, and eight, along with conservative one-sided
constraints on three others. Existing limits on the nine minimal operators at
dimension four are improved by factors of up to a billion, while 74 of our
explicit limits represent stringent first constraints on nonminimal operators.
Prospects are discussed for future analyses incorporating effects of Lorentz
violation in the matter sector, the role of gravitational Cherenkov radiation
by high-energy photons, data from gravitational-wave observatories, the
tired-light effect, and electromagnetic Cherenkov radiation by gravitons.
