There are many different ways (certainly lots of them are more elegant than this one):
matrix = RandomReal[1, {6, 6}];
DeleteCases[Table[If[EvenQ[i + j], matrix[[i, j]]], {i, 1, 6}, {j, 1, 6}], Null,Infinity] // MatrixForm
Here's another option:
Table[Reap[Do[If[EvenQ[i + j], Sow[matrix[[i, j]]]], {j, 1, 6}]], {i, 1, 6}][[All, 2, 1]] // MatrixForm
This one might be the nicest of the three:
Pick[matrix, Table[If[EvenQ[i + j], 1, 0], {i, 1, 6}, {j, 1, 6}], 1]
Cheers,
Marco