Paul,
The fibre-like characteristics present in some diffraction patterns of nucleic acids, which I think you are referring to, is the contribution of the helical transform of the double-stranded helix. Essentially, the length of the regular spacings in the base pairs modulates the intensities of the Bragg reflections. These then do not decay smoothly as the function of resolution which results in intensities along the helical axes at 3.4A being quite strong whilst perpendicular ones are much weaker. I have observed this in the diffraction patterns of RNA crystals but I have not seen the diffuse streaking along lattice lines that you show.
One recent example of fibre-like diffraction of a nucleic acid is illustrated here:
https://drive.google.com/file/d/0B-wXJqcqFfUHUVpvZDhqREdqMzA/edit?usp=sharing
I do not think that what you are seeing in your crystals can be attributed solely to the above. The streaking between the spots is probably some sort of lattice anomaly. It would be interesting to hear if this had impacted in some way on phasing or refinement of the structure.