I think of dimension (and texture) as one of the tools in the design toolbox, along with proximity, alignment, proportion, color, shape, and others. These tools allow us to create hierarchy, affordance, focus/highlights, contrast, legibility and other design paradigms that hopefully result in usable, visually pleasing design.
The effectiveness of each tool depends on the design objective (using color or proportion to highlight an element might be more effective than simply using alignment). In a minimalist approach, I like to work with the tools in "layers", in a specific order, and in quantities as small as possible. First, I begin with alignment and proximity — positioning elements. The challenge is to create something usable and attractive using only those tools. Sometimes, it's just not possible, so I'll move on to proportion, then color, then shape. I generally find that dimension and texture are unnecessary, so I leave them in the toolbox. Sometimes, perhaps because there's a high density of elements in the design, one element (button, dialogue, header) may need additional contrast or focus, at which point I'll consider adding some depth.