Compute $\sqrt{(31)(30)(29)(28)+1}$.
Note that the four numbers to multiply are symmetric with the center at $29.5$. 
Multiply the symmetric pairs to get $31\cdot 28=868$ and $30\cdot 29=870$.
$\sqrt{868\cdot 870 + 1} = \sqrt{(869-1)(869+1) + 1} = \sqrt{869^2 - 1^2 + 1} = \sqrt{869^2} = \boxed{869}$.