I suggest using triangles in pairs, since diamonds form a grid nicely.

5 large strips (with 4 macro-triangles each) can form an icosahedron in a fairly sane way.

But IMO the biggest mistake people make is trying to make everything fit on a single square; multi-tile objects are very useful. And at that point, why not make everything take several tiles?

Abandoning tiles entirely in favor of node adjacency can cut memory a lot but requires more thought.

I don't know if this is the real historical reason, but if you're doing something 2.5D, isometric, at with 2D graphics, at anything other than a 45 degree angle, then anything larger than one square creates clipping problems because part of it should either be behind another sprite on a square whose closest vertex is closer to the camera than the furthest vertex of the forward element. Z-ordering things on the ground between those elements gets even trickier. Making each building (or part of a building) stay within one square is by far the easiest way out of that predicament.