This is an interesting problem. I think a chess board is better expressed as a table than as a series of divs, as a screen reader would dictate the rows and columns where the figures are located. With a table:
table tr:nth-child(odd) td:nth-child(even) {
background: #000;
}
table tr:nth-child(even) td:nth-child(odd) {
background: #000;
}