This paper describes a fiber-shaped cellular building unit, named "cell fiber," for the construction of spatially-ordered 3D macroscopic tissues. The cell fiber, where the core of the fiber was cell-suspended extracellular matrix (ECM) hydrogel and the shell was Ca-alginate hydrogel, was continuously formed in a double co-axial laminar flow microfluidic device. Using the cell fibers, we demonstrate a fiber-based centimeter-scale 3D tissue constructs woven by our microfluidic weaving machine. We believe that our approach would be a powerful method for constructing macroscopic functional tissues with network-like spatially-ordered internal structures.