Does a black body (imagine like a square of solid black-body-matter) thermally conduct well at room temperature?
If you are talking of "black body" as in "black body radiation" better read the link:
"idealized" is the crucial point here. All matter can be approximated to a "black body", i.e an emitter of black body radiation by the multiplication with a constant specific for the material, the emissivity.
A black body will have a specific modified black body spectrum of radiation but can have any type of conductivity.For example dull wrought iron and black concrete have the same emissivity , .94, but very different thermal conductivity.