Given a 3-uniform hypergraph H and a positive integer k, the k-colored Ramsey numberR(H;k) is the minimum integer n such that every k-edge-coloring of the complete 3-uniform hypergraph Kn3 contains a monochromatic copy of H. Given two 3-uniform hypergraphs H and G, the constrained Ramsey numberf(H,G) is defined as the minimum integer n, such that any edge-coloring of Kn3 with any number of colors contains either a monochromatic copy of H or a rainbow copy of G. Let P33, C33 and S33 be the 3-uniform loose path, loose cycle and loose star of size 3, respectively. We study the constrained Ramsey number for the cases H,G∈{P33,C33,S33}. For 3-graphs H,G∈{P33,C33,S33}, we reduce f(H,G) to the 2-colored Ramsey number of H, except when H=G=S33, in which case we prove that f(S33,S33)=R(S33;2)+1. We also prove that f(H,P33)=R(H;2) for infinitely many 3-uniform hypergraphs H.