If you use \ limits_{down}^{up} you can get the text 'down' and 'up' above and below the integral signs. Also works for summation symbols.
[tex]\iiint \limits_{x \geq 3, y \geq 4, z \geq 5, 2z - x \geq 5} f(x, y, z)\, dx\,dy\,dz[/tex]
But it still looks crappy.
I'd advice just writing
[tex]\iiint \limits_{R} f(x, y, z)\, dx\,dy\,dz[/tex]
where
[tex]R=\{(x,y,z)|x \geq 3, y \geq 4, z \geq 5, 2z - x \geq 5\}[/tex]