This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functional. 

Let $Q_n=[-1/2,1/2]^n$ be the unit cube in $\mathbb R^n$ and let $[\xi^\perp]$ denote the hyperplane orthogonal to the vector $\xi\in S^{n-1}$. Then 
$$Vol_{n-1}(Q_n\cap [\xi^\perp])=\frac{1}{\pi}\int_{-\infty}^{\infty}
\prod\limits_{k=1}^n\frac{\sin(r \xi_k)}{r\xi_k}dr.$$

In our case $\xi=\xi_*=n^{-1/2}(1,1,\dots,1)$ so the integral becomes
$$Vol_{n-1}(Q_n\cap [\xi_*^\perp])=\frac{1}{\pi}\int_{-\infty}^{\infty}
\left(\frac{\sin(r/\sqrt n)}{r/\sqrt n}\right)^ndr.$$
In fact, the latter formula was already known to Laplace! The integral tends to $\sqrt {6/\pi}$, as $n\to\infty$ (e.g. by Laplace's method).  This can be also justified via the probabilistic interpretation suggested by Michael Lugo.


The Fourier analytic approach to sections of convex bodies is nicely presented in *The Interface between Convex Geometry and Harmonic Analysis* by A. Koldobsky and V. Yaskin. The derivation of the formula for volumes is available [here][1].


EDIT (15.01.2011). In fact both integrals can be calculated explicitly. The sinc integrals were studied by Borwein & Borwein (see also [*Multi-Variable sinc Integrals and the Volumes of Polyhedra*][2]).

For any $n\in \mathbb N$, $n>1$, we have
$$\frac{1}{\pi}\int_{-\infty}^{\infty}
\left(\frac{\sin(r/\sqrt n)}{r/\sqrt n}\right)^ndr =\frac{\sqrt n}{2^{n-1}(n-1)!}\sum_{k=0}^{n/2}(-1)^k{n \choose k}(n-2k)^{n-1}$$


  [1]: http://www.ams.org/bookstore/pspdf/cbms-108-prev.pdf
  [2]: http://algo.inria.fr/seminars/sem01-02/borwein1.pdf