How do you solve a net ionic equation without a balanced equation?

In class, my professor went over a method for finding the net ionic equation without determining the balanced equation (e.g., if finding the balanced equation was too difficult for us students). It involved writing the ions of each of the solutions being mixed and determining which ions would be spectator ions.

For example, finding the net ionic equation for the following mixture of solutions:

$\ce{NaC_2H_3O_3 + BaCl_2 +NH_4NO_3 + K_2CrO_4}$

Unfortunately, I don't quite grasp the process. How does one determine which ions and compounds will remain in the net ionic equation?

For this specific example, salts with alkali metal and ammonium cations are soluble. Nitrate and acetate anion salts are soluble. Therefore, all of those end up as spectator ions and we're left with just $\ce{Ba^2+}$ and $\ce{CrO_4^2-}$. Chromate salts other than those of alkali metal cations or ammonia are insoluble (rule 8 in the link), so the net ionic equation is:
$$\ce{Ba^2+_{(aq)} + CrO_4^{2-}_{(aq)} -> BaCrO_{4(s)}}$$