If you really insist on that form of output, here is a hack using obscure typesetting commands:
HoldForm[DisplayForm@RadicalBox[1, 3]] ==
DisplayForm@
Cell[BoxData[
RowBox[{"{",
Grid[{z} /. ComplexExpand[Solve[z^3 == 1, z]],
Alignment -> Left]}]], SpanMaxSize -> Infinity]
I wonder if it can be done more easily...
I would like to insert a TraditionalForm, but anybody knows where?