jmc

We first think of the Cubs forming a single block, denoted C, the Red Sox forming a single block R, and the Yankees forming a single block Y. Then there are $3!=6$ ways to arrange the three blocks in a row: $$\text{CRY, CYR, RCY, RYC, YRC, YCR.}$$Within each block, there are $3!$ ways to arrange the Cubs, $3!$ ways to arrange the Red Sox, and $2!$ ways to arrange the Yankees. Therefore, there are $$3!\times 3!\times 3!\times 2!=6\times 6\times 6\times 2=\boxed{432}$$ways to seat all eight All-Stars.