One illustrative method disclosed herein includes, among other things, defining a cavity in a plurality of layers of material positioned above a bottom source/drain (S/D) layer of semiconductor material, wherein a portion of the bottom source/drain (S/D) layer of semiconductor material is exposed at the bottom of the cavity, and performing at least one epi deposition process to form a vertically oriented channel semiconductor structure on the bottom source/drain (S/D) layer of semiconductor material and in the cavity and a top source/drain (S/D) layer of semiconductor material above the vertically oriented channel semiconductor structure. In this example, the method further includes removing at least one of the plurality of layers of material to thereby expose an outer perimeter surface of the vertically oriented channel semiconductor structure and forming a gate structure around the vertically oriented channel semiconductor structure.