Casework on $x$ ?

Solving for $y$ , we get that $3y = 2016-2z-6x$ . Since the RHS is even, the LHS must also be even, so $y$ must be even, i.e. $y=2k$ for some integer $k$ .

Solving for $z$ , we get that $2z = 2016-3y-6x$ . Since the RHS is divisible by 3, the LHS must also be divisible by 3, so $z$ must be divisible by 3, i.e. $z=3n$ for some integer $n$ .

Substituting this into the original equation, we get $6x+6k+6n = 2016$ , or $x+k+n=336$ . By stars and bars, this has a total of ${338 \choose 2} = 56953$ solutions. Therefore, our answer is $\boxed{56953}$ .

$6x+3y+2z=2016\\6x+3y=2016-2z\\2x+y=672-\frac{2}{3}z\\\text{Observe that z can range from 0 to 1008. Also observe that the RHS is always even}\\\text{, so there will always be a solution to this diophantine equation.}\\\text{The number of solutions will thus be } \frac{672-\frac{2}{3}z}{2}+1=337-\frac{1}{3}z\text{ for each value of z.}\\\text{Now let z=3k, so the answer is }\sum_{k=0}^{336}337-k=56953.$