My grandmother gave me a start of a plant several years ago, but never
knew what it was. It is small, doesn't have leaves, doesn't bloom,
and slowly spreads on its own. To make a new plant, just cut a few 1"
pieces off and stick them in soil.

I took three pictures that somebody can hopefully identify:

http://www.dnecentral.com/djw/tmp/plant1.JPG
http://www.dnecentral.com/djw/tmp/plant2.JPG
http://www.dnecentral.com/djw/tmp/plant3.JPG

Thanks!