A correct solution was also received from Philippe Fondanaiche. Two partial solutions were received.

Nathan Pauli provided the general solution. Here it is for the case of a 6 by 6 checkerboard.

The key observation is that if a domino straddles a line, then, somewhere else along that line, you must find another straddler. To see this, observe that if there were only one straddler across a line, then in each row (or column) along that line would there an odd number squares to be filled with dominoes at right angles to the lonesome straddler, a clear impossibility. So straddlers must come in pairs. This means that there are at most 6²/4 = 9 lines which can be straddled. But there are 10 lines that must be straddled!

The same argument for a larger size checkerboard fails, which leads one to believe that any larger checkerboard can have all its lines straddled. This is, in fact, the case! I'll leave it to you to find a general configuration.

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