One illustrative method disclosed herein includes, among other things, forming a liner semiconductor material within a trench, the liner material defining a transistor cavity, and forming spaced-apart source/drain placeholder structures that are at least partially positioned within the transistor cavity, the spaced-apart source/drain placeholder structures defining a gate cavity therebetween where a portion of the liner semiconductor material is exposed within the gate cavity. The method further includes forming a gate structure within the gate cavity and, after forming the gate structure, removing at least a portion of the source/drain placeholder structures to define a plurality of source/drain cavities within the transistor cavity on opposite sides of the gate structure, and forming a source/drain structure in each of the source drain cavities.