Well... many mathematicians, mainly the ones who works with topology, can think about spaces with more than 3 spatial dimensions... even me, who am not a mathematician, can imagine (only imagine, not visualize) spaces with 4 spatial dimensions... and its trippy... i wonder how they can think about spaces with 5, 6, or more spatial dimensions... damn...
Anyway, the 3D restriction to our usual thought is only a learned one. Our mind can think about several spatial dimensions, not only 3, but in the everyday world we only use 3 dimensions, so our minds become conditioned to think only in 3d, but it does not mean we can not think in greater dimensions... we only lack the practice of doing it. Also, as the OP mentioned, some psychedelics or meditation can allow some people to visualize higher dimensional spaces.

PS. Zebedee have you ever read some Castaneda's book?