Angular momentum $L = I\omega = mvr$ (for a particle)

$[L] = [m][v][r] = M \cdot LT^{-1} \cdot L = ML^2T^{-1}$

This is the same as the dimension of Planck's constant $h$, which makes sense since $E = h\nu$ and angular momentum appears in quantum mechanics through $\hbar = h/2\pi$.