Just to be clear, are you trying to move away from the minecraft style to a more traditional landscape renderer? The reason I ask is that on your link you say "taking inspiration from Voxel Farm and Minecraft".
Yup, that's right. I already have cubes, but I want to get rid of the blocky look and produce a smooth triangle mesh. I'll still be taking inspiration from Minecraft, though, as well as Voxel Farm. The renderer isn't specialized for voxels (anymore), it's just rendering triangles. I just want those triangles to be more smooth, instead of staircasing everywhere.
I haven't tried this out, but my current plan is to keep the uniform grid (for terrain generation), and sample the heightmap at the center of each xz-square, as well as at each corner, then create a "square-based pyramid", but making the height of each vertex correspond to the heightmap at that xz coordinate.