Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?

Gröbner bases come to mind, but for example, $$I = \langle x-y-z, y^4z^2+2y^3z^3+y^2z^4 \rangle = \langle x-y-z,x^2y^2z^2\rangle$$ has $$\left\{x^4y^2-2x^3y^3+x^2y^4, y^4z^2+2y^3z^3+y^2z^4, x^4z^2-2x^3z^3+x^2z^4, x-y-z\right\} $$ as its universal Gröbner basis, which makes me pessimistic about Gröbner methods. In this example, the symmetry can be broken by computing a primary decomposition which reveals the monomials, but what can be said in general?