Using Select with a function on 3d Table


From the following 3d-Table

t1={{{1, 1}, {2, 1}, {3, 4/3}}, {{1, 0}, {2, 1/2}, {3, 1}}, {{1, -1}, {2, 0}, {3, 2/3}}} 

I want to use Select for getting all pairs where the second Part is an Integer. So that I would get

{{{1, 1}, {2, 1}}, {{1, 0}, {3, 1}}, {{1, -1}, {2, 0}}} 

I can do it by

Pick[t1, Map[IntegerQ /@ #[[All, 2]] &, t1]] 

But there should be more elegant ways to do it. How to use Select or Select with Function or some other twist?