jmc

There are $3$ possibilities for the meat and $4$ possibilites for the dessert, for a total of $4\times 3=12$ possibilities for the meat and the dessert. There are $4$ possibilities for the first vegetable and $3$ possibilities for the second, but order doesn't matter, so we overcounted by a factor of $2$. For example, we counted 'baked beans and corn' and 'corn and baked beans' as $2$ different possibilities, so the total possibilites for the two vegetables is $\frac{4\times 3}{2}=6$, and the total number of possibilites is $12\times 6=\boxed{72}.$