Dispatched with an inoperative ACARS; unidentified air carrier flight crew failed to receive a message from dispatch making a required change to their destination alternate.

Narrative

We were flying an aircraft without ACARS. After landing; the Dispatcher informed us that they were not able to reach us via the commercial radio link while we were enroute to inform us of a required change in our destination alternate. After leaving 20;000 FT on departure; we contacted the departure airport ground station using radio #2 with our departure times; then started using radio #2 to monitor the commercial radio frequencies. Approaching Lake Michigan; I tuned in the ATIS frequency in an effort to start monitoring for an arrival ATIS. Once the ATIS was copied; I tuned the standby frequency in radio #2 to the ground frequency in order to make an in range call later in the descent. Unfortunately; the workload in the cockpit was a little higher than normal due to off course vectors and multiple step-downs due to arrival traffic. Ultimately I did not make the in range call; which would have given dispatch the opportunity to change our alternate at that time. I believe that dispatch tried to get a hold of us while I was off frequency to find out which runways were in use.