Whether nearest-neighbor is worth using depends on whether you personally like the the quality of the result that it gives. People will use smoother interpolation methods like linear and cubic when they desire/require smoother results.
As for accelerating things, I wouldn't expect that your own implementation would be faster, if done in MCode. INTERP3 uses optimized compiled C code specifically with the idea of helping things go fast. You could browse the File Exchange for customized interpolation code offered by other contributors,
As another suggestion, if you are making repeated calls to interp3 on the same volume array, it will be more optimal to pre-construct on interpolation operator using griddedInterpolant(). This lets you skip argument parsing and setup steps that interp3 might be repeating unnecessarily.
