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]}]