Not an answer, but too long for a comment: Mathematica is able to reduce the three-fold integral to a single integral, which does not seem to have a closed form answer:
$$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz=$$
$$\qquad =8\pi^2+16\pi\int_{0}^{\pi/2}\left[ \sqrt{ (2-\cos x)\cos x}-\cos x  \arcsin (1-\cos x)\right]\,dx$$

This evaluates to 135, while a numerical evaluation of the three-fold integral gives 248, so I presume the Mathematica output is not to be trusted.    
<sub>incidentally, 248 is remarkably close to $(2\pi)^3$.</sub>