Although centrosomes serve to organize microtubules in most cell types, oocyte spindles form and mediate meiotic chromosome segregation in their absence. Here, we used high-resolution imaging of both bipolar and experimentally generated monopolar spindles in Caenorhabditis elegans to reveal a surprising organization of microtubules and chromosomes within acentrosomal structures. We found that homologous chromosome pairs (bivalents) are surrounded by microtubule bundles running along their sides, whereas microtubule density is extremely low at chromosome ends despite a high concentration of kinetochore proteins at those regions. Furthermore, we found that the chromokinesin KLP-19 (kinesin-like protein 19) is targeted to a ring around the centre of each bivalent and provides a polar ejection force that is required for congression. Together, these observations create a new picture of chromosome-microtubule association in acentrosomal spindles and reveal a mechanism by which metaphase alignment can be achieved using this organization. Specifically, we propose that ensheathment by lateral microtubule bundles places spatial constraints on the chromosomes, thereby promoting biorientation, and that localized motors mediate movement along these bundles, thereby promoting alignment.