Following Gerald Edgar, 
$$W_3(1)\equiv\mathbb{E}[|Y|]=\int dx\int dy \,\sqrt{x^2+y^2}\,p(x,y)=2\pi\int_0^\infty r^2p(r)dr$$
with $p(x,y)dxdy=p(r)\,rdrd\phi$ the rotationally invariant distribution of the complex variable $Y=x+iy=r\cos\phi+ir\sin\phi$.    
From here
$$\mathbb{E}[|X|]=\int dx\int dy \,|x|\,p(x,y)=\int_0^\infty \int_0^{2\pi} |\cos\phi|r^2 p(r)\,drd\phi=\frac{2}{\pi}W_3(1).$$
According to equation (12) of <A HREF="https://www.carma.newcastle.edu.au/jon/walks2.pdf">Borwein, Straub, and Wan</A> the numbers $W_n(1)$ are given by
$$W_1(1)=1,\;\;W_n(1)=n\int_0^\infty J_1(x)J_0(x)^{n-1}\frac{dx}{x},\;\;n\geq 2.$$
We thus arrive at 
$$\int\limits_0^{2\pi}\int\limits_0^{2\pi}\int\limits_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz=(2\pi)^3\mathbb{E}[|X|]=16\pi^2 W_3(1)$$
$$\qquad=48\pi^2\int_0^\infty J_1(x)J_0(x)^{2}\frac{dx}{x}=3\;\frac{2^{1/3}}{\pi^2}\Gamma({\textstyle \frac{1}{3}})^6
+
108\;\frac{2^{2/3}}{\pi^2}\Gamma({\textstyle \frac{2}{3}})^6 \approx 248.65$$

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comments:

- since $(2\pi)^3\approx 248.05$, the average of the absolute value of the sum of three cosines equals unity within one-quarter of a percent.
- the $n$-fold integral 
$$I_n\equiv\int_0^{2\pi}\cdots\int_0^{2\pi}|\cos x_1+\cdots + \cos x_n|\,dx_1\cdots dx_n=2^{n+1}\pi^{n-1}n\int_0^\infty J_1(x)J_0(x)^{n-1}\frac{dx}{x}$$ 
equals a power of 2 for $n=1$ and $n=2$, but not for $n=3$ (nor for larger $n$, as far as I have checked).
- numerics suggests that $(2\pi)^{-n}I_n\propto\sqrt n$ for large $n$, but I have not been able to derive this asymptotics from the Bessel-function integral expression.