An idea for a fabric that is 3D printable on an Ultimaker, a MakerBot, a Reprap, or a device at least as capable. Highly experimental!
Prototype modeled in Rhino 3D.
Update: Someone pointed me to a fabric design that has been proven to be printable - great work! It is: http://www.thingiverse.com/thing:8724
Step 1: Print the solids.
Step 2: Print the loops which function as connectors. Every loop is a single thick plastic thread, to be printed in one go! Experiment with how much the thread will settle during printing. Adapt the input data accordingly. I do not own a 3D printer, and I have no idea whether this is feasible.
Optionally push down the loops after printing is done.
Step 3: Print the 3D threads that secure the connectors.