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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: $$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz=$$ $$\qquad =8\pi^2+16\pi\int_{0}^{\pi/2}\left[ \sqrt{ (2-\cos x)\cos x}-\cos x \arcsin (1-\cos x)\right]\,dx$$