We develop a conceptually clear, intuitive, and feasible decision procedure for testing satisfiability in the full multi\-agent epistemic logic \CMAELCD\ with operators for common and distributed knowledge for all coalitions of agents mentioned in the language. To that end, we introduce Hintikka structures for \CMAELCD\ and prove that satisfiability in such structures is equivalent to satisfiability in standard models. Using that result, we design an incremental tableau-building procedure that eventually constructs a satisfying Hintikka structure for every satisfiable input set of (...) formulae of \CMAELCD\ and closes for every unsatisfiable input set of formulae. (shrink)