$printCache = {};
cachedPrint[code_] :=
  Internal`InheritedBlock[{Print},
   Unprotect@Print;
   Print[stuff__] /; (! TrueQ@$printCachingOverload) :=

    Block[{$printCachingOverload = True},
     AppendTo[$printCache, {stuff}];
     Print[stuff]
     ];
   Protect@Print;
   code
   ];
cachedPrint~SetAttributes~HoldFirst;

cachedPrint[Do[someSuperCoolFunction[x]; Print[x, " etc."], {x, 10}]]

In[33]:= $printCache

Out[33]= {{1, " etc."}, {2, " etc."}, {3, " etc."}, {4, " etc."}, {5,
  " etc."}, {6, " etc."}, {7, " etc."}, {8, " etc."}, {9,
  " etc."}, {10, " etc."}}

In[34]:= DownValues@Print

Out[34]= {}