You can use Surd to also get the negative real root:
Solve[-3 Surd[(y - 2)^4, 5] + 29 == -19, y]
(* {{y -> -30}, {y -> 34}} *)
As an alternative the equation can be rewritten:
Solve[-3 ((y - 2)^4)^(1/5) + 29 == -19, y, Integers]
(* {{y -> -30}, {y -> 34}} *)