We generalize the string-net construction to multiple flavors of strings, each of which is labeled by the elements of an abelian group $G_i$. The same flavor of strings can branch while different flavors of strings can cross one another and thus they form intersecting string-nets. We systematically construct the exactly soluble lattice Hamiltonians and the ground state wave functions for the intersecting string-net condensed phases. We analyze the braiding statistics of the low energy quasiparticle excitations and find that our model can realize all the topological phases as the string-net model with group $G=\prod_i G_i$. In this respect, our construction suggests several ways of building lattice models which realize topological order $G$. They correspond to intersecting string-net models with various choices of flavors of strings associated with different decomposition of $G$. In fact, our construction concretely demonstrates the K\text{ü}nneth formula by constructing various lattice models with the same topological order. As an example, we construct the $G=\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}_2 $ string-net model which realizes a non-abelian topological phase by properly intersecting three copies of toric codes.