Since the lattice spacing is about eight angstroms, the issue isn't any sort of unusual lattice spacing. Instead, the issue is that bones are thick.
[Another answer points out that the crystals are also pretty small, but I don't think that invalidates the discussion below.]
[By NASA/JPL-Caltech/Ames - http://photojournal.jpl.nasa.gov/jpeg/PIA16217.jpg, Public Domain, Link]
In x-ray diffraction (and other diffraction) experiments, the most common diffraction angle is $0^\circ$, which is sometimes called "forward scattering." In most x-ray diffraction images, there's a very bright spot in the forward beam direction.
For example in the image above, there are some red pixels near the dark semicircle at the bottom of the figure; that's the forward-scattered (or unscattered) part of the x-ray beam. The bright rings tell you the angles at which light is Bragg-scattered from various crystal planes; they are rings, rather than points, because the material here (a martian rock) was made of many crystals, rather than just one.
Now think about those diffracted x-rays. If they're traveling through more of your material (in your case, bone), they have some chance to be diffracted again by interacting with another part of the crystal. So the way to get a nice image like the one above is to send your x-rays through a sample that is relatively thin. There's a little bit of an art to it: the thinner the sample, the cleaner the diffraction lines are, but the less of the incident beam actually gets diffracted. Thicker samples give brighter diffraction patterns, but the patterns are messier due to multiple diffraction.
A medical x-ray of your skeleton actually shows something totally different from a diffraction image like the one above. The bones are thick enough to absorb the x-rays that pass through them --- or at least to scatter all of the x-rays away from the forward beam direction. What's recorded on the x-ray film are dark places where x-rays do hit, and brighter places where x-rays don't hit: the bright places on the x-ray film are the shadows of the dense parts of the subject. (Photographers call this a negative image.) Any x-rays diffracted away from the bones are most likely to diffract into a region of the film that's directly exposed to the primary x-ray beam, where they'll be overwhelmed.
The fact that bone is white, and the shadows on x-ray negatives are white, is just a concidence; those left-right markers aren't really white, but they cast a shadow just the same.