This question is a special case of Deligne-Simpson Problem which asks about restrictions on conjugacy classes of square matrices $A_1,...,A_k$ whose product equals $1$. See for instance
http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_06.pdf
and references therein.
(And, yes, there are some nontrivial restrictions even for triples of matrices!)
There is also an interesting variation on this question where one is asking about singular values of the matrices (google "Thompson's conjecture").