There is a positive answer to this question now by Pham Hung Quy's post here and this paper, 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 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.

There is a positive answer to this question now by Pham Hung Quy's post here and this paper, 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.