  We study Ising antiferromagnets that have nearest-neighbour interactions on
multilayer triangular lattices with frustrated ($abc$ and $abab$) stacking, and
make comparisons with the unfrustrated ($aaa$) stacking. If interlayer
couplings are much weaker than in-plane ones, the paramagnetic phase of models
with frustrated stackings has a classical spin-liquid regime at low
temperature, in which correlations are strong both within and between planes,
but there is no long-range order. We investigate this regime using Monte Carlo
simulations and by mapping the spin models to coupled height models, which are
treated using renormalisation group methods and an analysis of the effects of
vortex excitations. The classical spin-liquid regime is parametrically wide at
small interlayer coupling in models with frustrated stackings. By contrast, for
the unfrustrated stacking there is no extended regime in which interlayer
correlations are strong without three-dimensional order.
