jmc

For a quadratic to have two real roots, the discriminant must be greater than 0. So we require $$\begin{array}{rl}{7}^2-4\cdot 3\cdot c& >0\Rightarrow \\ 49-12c& >0\Rightarrow \\ c& <\frac{49}{12}.\end{array}$$The largest integer smaller than $\frac{49}{12}$ is 4. Thus, the positive integer values of $c$ are 1, 2, 3, and 4, and their product is $\boxed{24}$.