How do I connect my Acoustimass® 5 Series III

Well, first of all this speaker system is designed to give you a subwoofer where you do not have a sub-woofer output on your receiver/amp. However, that is not the case with the Onkyo. The way I would set it up (only because of the speakers you have) according to the Acoustimass Manual. You take the left and Right "Speaker" outputs to the Left and Right input on the Acoustimass and the speakers crossover does the rest. The Onkyo is designed to have a powered subwoofer so in this format you would just tell the receiver to turn off the sub channel which would tell the Onkyo to send the low end to LR. That should do it.