The transport of parton polarization in strong interactions is of fundamental interest. It is an open question whether the polarization of quarks or antiquarks produced in hard collisions is obesrvable via the final-state fragmentation products in the resulting jets. We have searched for signatures of polarization in hadronic jets from $Z_{0} \rightarrow q\bar{q}$ decays using the ``jet handedness'' method. The polar angle asymmetry induced by the high SLAC Linear Collider electron-beam polarization was used to separate quark jets from antiquark jets, expected to be left and right polarized, respectrively. We find no evidence for a nonzero jet handedness in our global sample or in a sample of light quark jets, implying that the transport of polarization through the quark fragmentation process is small.