There are two main forms of Icehouse pieces that can made from paper: 3-D and 2-D. The 3-D pieces were initially produced and marketed by Icehouse Games, Inc. as "Origami Icehouse". The 2-D pieces were conjured up by game enthusiast Morgan Andrews in early 2007, inspired by the depiction of Icehouse pieces in games played online at Super Duper Games, though they are similar to Zarf's Zarcana Pieces which are painted on wood (like the Mona Lisa).
Three-dimensional paper pieces
For the industrious and/or cheap, Looney Labs has available Paper Icehouse, a set of perforated sheets of cardstock, three to a sheet -- one of each size. When punched out, folded, and taped or glued shut, the set yields a full standard Icehouse set. That is, you make a stash of red, a stash of yellow, a stash of green, and a stash of blue. Note that these pieces are not stackable (unless you clip off their bottoms).
Looney Labs has discontinued the paper pyramids. However you can fold your own.
Two-dimensional paper pieces
A quicker, easier way to make a stash of pieces for playing some Icehouse games is to draw them on flat squares of cardstock or paper. These are useful for players who don't have their 3-D pices with them at times when they want to play Icehouse games.
To construct a stash of 2-D pieces: Cut cardstock into 15 square slips, (5 half-inch, 5 three quarters of and inch, and 5 one-inch). Draw 5 small, 5 medium and 5 large triangles on the slips, mark them with 1, 2 or 3 dots respectively, and color them the desired color of the stash. These represent the pieces in their "flat" position. On the flip side of the slips, do the same thing bu with squares instead of triangles. These represent the pieces in their "standing" position. Though these flat Icehouse pieces are not ideal for stacking, they can be easily used to play many games, including Homeworlds, Pikemen and more.