Grid[Table[Style[1, Opacity[0]], {8}, {8}], Frame -> All,
Background -> {{}, {},
Flatten@Table[
If[EvenQ[i + j], {i, j} -> Red, {i, j} -> Black], {i, 8}, {j,
8}]}]
Another code, the same result:
Grid[Table[Style[1, Opacity[0]], {8}, {8}], Frame -> All,
Background -> {{}, {},
If[EvenQ[Total[#]], # -> Red, # -> Black] & /@
Flatten[Array[List, {8, 8}], 1]}]