We introduce the sum-rank metric analogue of Reed--Muller codes, which we called linearized Reed--Muller codes, using multivariate Ore polynomials. We study the parameters of these codes, compute their dimension and give a lower bound for their minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Reed--Muller codes in the Hamming metric. Finally, we also show that many of the newly introduced linearized Reed--Muller codes can be embedded in some linearized Algebraic Geometry codes, recently defined in arXiv:2303.08903, a property which could turn out to be useful in light of decoding.