Example 3 Find all $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying the following inequality: $f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+3 z)$, for all $x, y, z \in \mathbf{R}$.
For any $t \in \mathbf{R}$, let $x=t, y=z=0$, from the condition we know $f(t)+f(0)+f(t) \geqslant 3 f(t), f(0) \geqslant f(t)$; then let $x=y=\frac{t}{2}, z=-\frac{t}{2}$, we have $f(t)+f(0)+f(0) \geqslant 3 f(0), f(t) \geqslant f(0)$. Therefore, for any $t \in \mathbf{R}, f(t)=f(0)$, i.e., $f$ must be a constant. Conversely, any constant $f$ must satisfy the given condition.