There is a positive answer to this question now by Pham Hung Quy's post here and [this paper][1], but one can also ask this question about other gradings.  This is not an answer, but a long comment: 

 - $\mathbb{N}^n$-graded are monomial ideals: It is well-known that monomial irreducible decomposition is irreducible decomposition.  
 - Ideals finely graded (Hilbert function takes only the value 1) by an affine semigroup (a finitely generated subsemigroup of $\mathbb{Z}^n$) are toric ideals.  They are prime and irreducible.
 - Ideals graded by some affine semigroup $A$ could be an interesting next goal.  Since the grading is torsion free, there is an $A$-graded primary decomposition, but irreducible decomposition is not clear.

Descending further in the hierarchy of grading monoids, a *commutative Noetherian monoids* is any quotient of $\mathbb{N}^n$ by a congruence.  An ideal is graded by  such a monoid if and only if it is a binomial ideal (Proposition 1.11 in *Binomial Ideals* by Eisenbud and Sturmfels) and the theory of binomial ideals can be developed somewhat similarly to that of monomial ideals.  However, an ideal graded by a fixed commutative Noetherian monoid $Q$ need not even have a primary decomposition into $Q$-graded ideals.  For example, $(x^2-xy, xy-y^2) = (x,y^2) \cap (x-y)$.  The ideal on the left hand side is (finely) $Q$-graded for some monoid $Q$ in which $x$ and $y$ have different degrees, but they must have the same degree in the unique minimal prime $(x-y)$.

Now one could ask if in general binomial ideals have binomial irreducible decompositions, that is, letting $Q$ vary in the decomposition.  This was an open problem in Eisenbud/Sturmfels, but the answer is negative by Section 6 in [*Irreducible decomposition of binomial ideals* by Kahle, Miller, O'Neill][2].




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