How to use Mathematica to do a complex integrate with poles in real axis?

Try

Integrate[Exp[I z] 1/z, {z, -Infinity, Infinity},PrincipalValue -> True]
(*I π*)

One can also consider using the residue theorem. The residue is readily obtained by

Residue[Exp[I z] 1/z, {z, 0}]

returning 1, which means that the integral is $ \mathrm i \pi $.

If you are sure about your integral's behavior you can try

Integrate[Exp[I z] 1/z, {z, -Infinity, Infinity}, 
 GenerateConditions -> False]
(* I π *)