We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C *-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C *-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.