  Chiral effective field theory (EFT) enables a systematic description of
low-energy hadronic interactions with controlled theoretical uncertainties. For
strongly interacting systems, quantum Monte Carlo (QMC) methods provide some of
the most accurate solutions, but they require as input local potentials. We
have recently constructed local chiral nucleon-nucleon (NN) interactions up to
next-to-next-to-leading order (N$^2$LO). Chiral EFT naturally predicts
consistent many-body forces. In this paper, we consider the leading chiral
three-nucleon (3N) interactions in local form. These are included in auxiliary
field diffusion Monte Carlo (AFDMC) simulations. We present results for the
equation of state of neutron matter and for the energies and radii of neutron
drops. In particular, we study the regulator dependence at the Hartree-Fock
level and in AFDMC and find that present local regulators lead to less
repulsion from 3N forces compared to the usual nonlocal regulators.
