Yes - surface bundles over the circle are irreducible as long as the fiber is not a two-sphere.  This follows from the fact that the universal cover of such a surface bundle is homeomorphic to $\mathbb{R}^3$ and Proposition 1.6 of Hatcher's three-manifold [notes][1]. 


  [1]: http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html