Photons travel in 3D space and may only be occluded by 2D surfaces. There is no getting around that fundamental aspect of reality. In this universe.

This is why as an armchair physicist I'm excited about the new neutrino observations and research going on, mainly because neutrinos aren't affected by gravitational lensing and other effects other particles encounter as they 'roll around' on space-time. Would it be possible to understand conceptually how other dimensions extend out of space time? I'm imagining a Y-axis perpendicular to the space-time fabric....

Gravitational lensing is a distortion of spacetime. It affects all particles, including neutrinos.