After a little searching, I found this idea: printing using styrofoam from styrofoam picnic plates! This looked very interesting. Styrofoam plates are easy to find, at very little cost. Some styrofoam containers could also be repurposed for this, such as meat trays or fast food holders. Using a large styrofoam plate gives a pretty big flat area (the center) -- so the tilted edges can be cut off, and this flat area can be used for making a drawing.
Then paint can be added, by brushing some on with a paintbrush, and then a print can be made by either pressing paper onto the styro etching surface, or picking up the etched and painted styrofoam piece and pressing it onto paper.
As you can see, you'll get a mirror image -- so if you want to put words in the design, keep it in mind that you'll need to etch in a mirror image of the word (write each letter backwards, and write from right to left).
Each print made with this technique is a mono print; you can reapply paint to make more of the same, but if you try to make more than one without repainting you will probably not have a very vivid print. Here are some examples: the tulip on the left is a first print, and on the right is a second print made without repainting. Sometimes a kind of faded appearance does look nice; so you can experiment with doing reprints, if you want to achieve a lighter look.
So the next time you have some leftover styrofoam dishes or containers in the house, remember "styro-printing!" You might like to repurpose some containers and try this out; or you can easily find a package of styrofoam plates at the grocery store-- I was glad to find that this material was so easy to make designs in (with a blunt, rounded but kind of pointy instrument-- a knitting needle or crochet hook could also work for this; I think another time I'll try using those . . .)