  In the first part of the paper the natural scheme for proving noncommutative
individual ergodic theorems for multiple sequences is described and applied to
obtain results on unrestricted convergence of multiaverages. In the second part
convergence of ergodic averages induced by several maps satisfying specific
recurrence relations, including so-called Multi Free Group Partial Sums, is
proved. This is the multiindexed version of results obtained earlier jointly
with V.I.Chilin and S.Litvinov.
