The Bergman projection operator \(T_\beta \) , as is well known, continuously maps weighted Lebesgue and some more general spaces onto their holomorphic or harmonic subspaces. In the setting of the unit ball in \({\mathbb R}^n,\) it is shown that the result continues to hold for mixed norm spaces \(L(p,q,\alpha),\, \alpha >0\) , while for non-positive \(\alpha \le 0\) , the mixed norm space \(L(p,q,\alpha)\) has harmonic Besov space \(h\Lambda _\alpha ^{p,q}\) as an image under the Bergman projection \(T_\beta \) . On the other hand, \(T_\beta \) fails to be a bounded projection on the Bloch space. Another family of Bergman type operators \(\Phi \) is constructed whose members continuously project the three-parameter Besov space \(\Lambda _\alpha ^{p,q}\) onto its harmonic subspace \(h\Lambda _\alpha ^{p,q}\) as well as the Bloch space of smooth functions onto its harmonic subspace.