Using an ab initio method, we explored electronic structures and transport properties of zigzag graphene nanoribbons (ZGNRs) with ordered doping of B or N atoms. We find B or N atoms doping can increase significantly the conductance of the ZGNRs with an even number of zigzag chains due to additional conducting channels being induced and the breakdown of parity limitation. The higher the doping concentration, the larger the current amplification factor obtained. For the nanojunctions with one row B (or N) atoms, the current amplification factor can be larger when the doping position is near to the center, while for the junction with two rows, the trend is subtle due to the interactions between the two rows of B (or N) atoms. Negative differential resistive phenomena are found for the case of B doping at low concentrations and the case for N doping. The conductance of the ZGNR with odd numbers of zigzag chains can also be increased by doping of B or N atoms. More interestingly, the B or N doping can almost completely remove the even-odd effect on electronic transport of the ZGNRs. Our studies provide avenues to drastically improve the electronic transport of ZGNRs, helpful for graphene applications.