  We show that Majorana fermions (MFs) exist in two- and three-dimensional
(2D,3D) fermionic optical lattices with strictly 1D spin-orbit coupling (SOC)
which has already been realized in ex- periments. For a quasi-1D topological
BCS superfluid, there are multiple MFs at each end which are topologically
protected by a chiral symmetry. In the generalization to higher dimensions, the
multiple MFs form a zero energy flat band. An additional experimentally tunable
in-plane Zeeman field drives the system to a topological Fulde-Ferrell (FF)
superfluid phase. We find that even though the multiple MFs are robust against
the in-plane Zeeman field if the order parameters at the different chains are
enforced to be identical, they are destroyed in the self-consistently obtained
FF phase where the order parameters are inhomogeneous on the boundaries. Our
results are useful to guide the experimentalists on searching for MFs in the
context of ultracold fermionic atoms.
