High Frequency Transformer Design and Materials
I get the concept that ideally you want to transformer to not hold any
  magnetic energy and instead to transfer it all, but what material
  properties allow this?


This concept doesn't fit into my way of thinking. For a straight transformer with an AC voltage applied to the primary and a secondary on load, the ampere turns of the secondary (magneto motice force) is totally cancelled by the ampere turns in the primary that resulted from that secondary load. Should you disconnect the secondary load, the primary is just an inductor having an inductance determined by the core material, shape, gaps (if any) and number of turns.

To that end, for a HF transformer, you pick a ferrite material that has low losses at the operating frequency (read the material data sheets for this) and then you begin the process of determining the number of primary turns so as not to cause excessive saturation.

This means trialling an estimate of inductance, calculating the number of turns and therefore calculating the MMF (ampere turns) for the primary under no load conditions. You then factor in the mean magnetic field length (it's a core parameter contained in the data sheet) to calculate H: -

H = ampere turns per metre

Amps is based on inductance, frequency and AC voltage applied to primary just as any inductor would be. Take the peak current and multiply by primary turns and divide by effective core length.

Then go to the core parameters in the data sheet and see if the value of H is going to cause excessive saturation - i.e. use the BH curve.

If it looks like too much saturation then you'll need to increase the turns and possibly implement a gap. Same method as a flyback transformer.