The Rademacher sums are investigated in the Morrey spaces Mp,w on [0,1] for 1≤p<∞ and weight w being a quasi-concave function. They span l2 space in Mp,w if and only if the weight w is smaller than log2−1/22t on (0,1). Moreover, if 1<p<∞ the Rademacher subspace Rp,w is complemented in Mp,w if and only if it is isomorphic to l2. However, the Rademacher subspace R1,w is not complemented in M1,w for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces Mp,w is described. It turns out that for any infinite-dimensional subspace X of Rp,w the following alternative holds: either X is isomorphic to l2 or X contains a subspace which is isomorphic to c0 and is complemented in Rp,w.