To solve the integral $\int_0^1 \int_0^1 \int_0^1 ([x]+[y]+[z]) \, dx \, dy \, dz$, where $[x]$, $[y]$, and $[z]$ denote the greatest integer functions for $x$, $y$, and $z$ respectively, we first need to understand how the greatest integer function behaves over the interval from 0 to 1.

The greatest integer function, denoted by $[a]$, gives the greatest integer less than or equal to $a$. Hence, for any $x \in [0, 1)$, $[x] = 0$. This is similarly true for $y$ and $z$ as they are also integrated over the interval [0, 1).

Therefore, within the integration limits from 0 to 1, $[x] = 0$, $[y] = 0$, and $[z] = 0$ for all points in that range. Consequently, the expression inside the integral $( [x] + [y] + [z] )$ simplifies to 0:

$$\int_0^1 \int_0_1 \int_0^1 ([x] + [y] + [z]) \, dx \, dy \, dz = \int_0^1 \int_0_1 \int_0^1 (0) \, dx \, dy \, dz = 0.$$

Since the integrand is zero throughout the entire volume of integration, the value of the integral is indeed 0.

Thus, the correct answer is 0.