When $ \mathbf{a} + k \mathbf{b} $ and $ \mathbf{a} - k \mathbf{b} $ are orthogonal, their dot product is 0: \[(\mathbf{a} + k \mathbf{b}) \cdot (\mathbf{a} - k \mathbf{b}) = 0.\]Expanding, we get \[\mathbf{a} \cdot \mathbf{a} - k \mathbf{a} \cdot \mathbf{b} + k \mathbf{a} \cdot \mathbf{b} - k^2 \mathbf{b} \cdot \mathbf{b} = 0.\]Since $ \mathbf{a} \cdot \mathbf{a} = \|\mathbf{a}\|^2 = 9 $ and $ \mathbf{b} \cdot \mathbf{b} = \|\mathbf{b}\|^2 = 16, $ we are left with $ 9 - 16k^2 = 0 $. Then $ k^2 = \frac{9}{16}, $ so the possible values of $ k $ are $ \boxed{\frac{3}{4}, -\frac{3}{4}} $.