This is a variant of finding a total-degree based power series.
linearize[poly_] := Module[
{vars = Variables[poly], t},
Normal[Series[poly /. Thread[vars -> t*vars], {t, 0, 1}]] /. t -> 1
]
The example:
poly = 1 + x + y + z + x y + x y z;
linearize[poly]
(* Out[50]= 1 + x + y + z *)
