  We introduce a new method to propagate uncertainties in the beam shapes used
to measure the cosmic microwave background to cosmological parameters
determined from those measurements. The method, which we call Markov Chain Beam
Randomization, MCBR, randomly samples from a set of templates or functions that
describe the beam uncertainties. The method is much faster than direct
numerical integration over systematic `nuisance' parameters, and is not
restricted to simple, idealized cases as is analytic marginalization. It does
not assume the data are normally distributed, and does not require Gaussian
priors on the specific systematic uncertainties. We show that MCBR properly
accounts for and provides the marginalized errors of the parameters. The method
can be generalized and used to propagate any systematic uncertainties for which
a set of templates is available. We apply the method to the Planck satellite,
and consider future experiments. Beam measurement errors should have a small
effect on cosmological parameters as long as the beam fitting is performed
after removal of 1/f noise.
