Assuming that the three parties stick around...
Probably what would happen is that two of the three parties position themselves at an ideological extreme, with the third being moderate/in-between the two. If they were any closer, you'd see parties start to merge, and then you'd be back at a two-party system.
Probably the moderates wouldn't be very popular - I don't see them getting in the White House. However, if a party fails to gain enough seats to totally control the house or the senate, the moderates may be courted by either side for support for their various agendas. So, while they don't have as many seats, if the party sticks together they could have disproportionate influence in American politics.