A turbine shroud comprising: an arcuate wall having opposite back and front sides (30, 32), and forward and aft hooks (34, 36) extending from said back side at opposite ends thereof; and, multiple rows of film cooling holes inclined longitudinally through said wall, wherein overcrossing adjacent rows (1, 3; 2, 4) of said holes are inclined in opposite aft and forward directions for channeling cooling air therethrough in counter flow; the shroud further comprising: a pocket bounded forward and aft by said hooks (34,36) and bounded laterally by side rails; each of said holes having an inlet on said back side inside said pocket, and an outlet on said front side; and said inlets are spaced apart laterally and longitudinally inside said pocket, and said outlets are spaced apart laterally and longitudinally on said front side in different patterns; characterized in that: adjacent pairs of said film cooling holes in adjacent rows (1,3;2,4) of said multiple rows of film cooling holes are oppositely inclined longitudinally so that said adjacent pair of film cooling holes (1,3;2,4) converge together longitudinally as they extend from said back side to said front side. an arcuate wall having opposite back and front sides (30, 32), and forward and aft hooks (34, 36) extending from said back side at opposite ends thereof; and, multiple rows of film cooling holes inclined longitudinally through said wall, wherein overcrossing adjacent rows (1, 3; 2, 4) of said holes are inclined in opposite aft and forward directions for channeling cooling air therethrough in counter flow; the shroud further comprising: a pocket bounded forward and aft by said hooks (34,36) and bounded laterally by side rails; each of said holes having an inlet on said back side inside said pocket, and an outlet on said front side; and said inlets are spaced apart laterally and longitudinally inside said pocket, and said outlets are spaced apart laterally and longitudinally on said front side in different patterns; characterized in that: adjacent pairs of said film cooling holes in adjacent rows (1,3;2,4) of said multiple rows of film cooling holes are oppositely inclined longitudinally so that said adjacent pair of film cooling holes (1,3;2,4) converge together longitudinally as they extend from said back side to said front side. a pocket bounded forward and aft by said hooks (34,36) and bounded laterally by side rails; each of said holes having an inlet on said back side inside said pocket, and an outlet on said front side; and said inlets are spaced apart laterally and longitudinally inside said pocket, and said outlets are spaced apart laterally and longitudinally on said front side in different patterns; characterized in that: adjacent pairs of said film cooling holes in adjacent rows (1,3;2,4) of said multiple rows of film cooling holes are oppositely inclined longitudinally so that said adjacent pair of film cooling holes (1,3;2,4) converge together longitudinally as they extend from said back side to said front side. adjacent pairs of said film cooling holes in adjacent rows (1,3;2,4) of said multiple rows of film cooling holes are oppositely inclined longitudinally so that said adjacent pair of film cooling holes (1,3;2,4) converge together longitudinally as they extend from said back side to said front side.