There is a positive answer to this question now by Pham Hung Quy's post here and [this paper][1], but let me also mention that the same thing is not true if the $\mathbb{Z}$-grading is changed to a $Q$-grading where $Q$ is an arbitrary commutative Noetherian monoid. 

By a result of Eisenbud and Sturmfels (*Binomial ideals*, Proposition 1.11), $Q$-graded ideals are exactly binomial ideals.  There are binomial ideals that are reducible as ideals, but cannot be written as intersections of binomial ideals.  My [paper with Ezra Miller and Chris O'Neill][2] gives the first example: In $k[x,y]$, the ideal $(x^2y-xy^2, x^3, y^3)$ has two irreducible components (because its socle is two dimensional), and one of them must contain a non-binomial.  See Example 6.1 and Theorem 6.4 in the linked paper.


  [1]: http://arxiv.org/abs/1508.07518
  [2]: http://arxiv.org/abs/1503.02607