In this talk we shall discuss our recent work which shows that in the periodic homogenization of viscous HJ equations in any spatial dimension the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original Hamiltonian. Moreover, the loss of quasi convexity is, in a way, generic: when the spatial dimension is 1, every convex function can be modified on an arbitrarily small interval so that the modified function, G(p),  is quasiconvex, and for some Lipschitz continuous periodic V(x),  the  effective Hamiltonian corresponding to H(p,x)=G(p)+V(x) is not quasiconvex. This observation is in sharp contrast with the inviscid case where homogenization preserves quasiconvexity. The talk is based on joint work with Atilla Yilmaz, Temple University.