Summary: We define a combinatorial checkerboard to be a function \(f:\{1, \dots ,m\}^d\to\{ 1,-1\}\) of the form \(f(u_1,\dots,u_d)=\prod_{i=1}^df_i(u_i)\) for some functions \(f_i:\{1,\dots ,m\}\to\{ 1,-1\}\). This is a variant of combinatorial rectangles, which can be defined in the same way but using \(\{ 0,1\}\) instead of \(\{ 1,-1\}\). We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case \(m=2\). We construct a pseudorandom generator that \(\epsilon\)-fools all combinatorial checkerboards with seed length \(O\left(\log m+\log d\cdot\log\log d+\log^{3/2} \frac{1}{\epsilon}\right)\). Previous work by R. Impagliazzo, N. Nisan and A. Wigderson [“Pseudorandomness for network algorithms”, in: STOC ‘94. Proceedings of the 26th ACM Symposium on Theory of Computing. New York, NY, USA, 1994. New York, NY: ACM. 356–364 (1994; doi:10.1145/195058.195190)] implies a pseudorandom generator with seed length \(O\left(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\right)\). Our seed length is better except when \(\frac{1}{\epsilon}\geq d^{\omega(\log d)}\).