For thermoelectric conversion, I would estimate that Nantenna technology would be a good first step in converting x-rays to DC power.
http://en.wikipedia.org/wiki/Nantenna In the DGT reactor, there is a nickel foam support the is in the zone for maximum x-ray production. This foam can support 1 to 2 nanometer nantenna structures imbedded in the foam fibers that can receive this x-ray radiation and convert it to DC current. The nickel fibers can also support the DC current transport network to the outside of the reactor. The individual nantenna could be fashioned as a wide band fractal EMF antenna nanostructure with a maximum performance range for receiving x-rays in the wavelength size range that corresponds to the NAE. Because of the high energy content of x-rays and their corresponding blackbody temperature association, high Carnot efficiency in the 90% range might be possible.