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Wikipedia has a fairly good overview of this issue -- how and why a two-party system evolves, and what the pros and cons are.

Based on the analysis there, I would suggest that there are three pillars to two-party domination. A solution would have to address one or more of them:

1. Restrictions on who can be on the ballot. On the simplest level, this is an obvious one. The harder it is for a small party to get on the ballot, the less likely they will get on the ballot, the less likely they will be able to compete for seats. On a subtler level, the difficulty involved in getting on the ballot makes it much more likely that people will try to have influence within one of the major parties, rather than trying to establish power independently of them. So in the U.S., liberals are much more likely to stay within the Democratic Party and try to influence it's direction than they are to try to establish a viable Liberal Democratic Party. There are numerous factions within the major parties, rather than numerous parties.
2. A plurality is all that's needed to win. In countries where a majority is required, coalition building between parties is often necessary. Third parties can become kingmakers. Since only a plurality is necessary in the U.S., there is no need for such arrangements, and third parties don't have this leverage.
3. Proportional representation instead of independent districts. In countries where legislative seats are distributed based on total vote counts, a party getting a small percentage of the vote would still get a few seats. But our system of voting by district only gives seats to candidates who win a majority within that district.

I don't believe that any of these pillars are built into the U.S. Constitution; rather, they are long-standing traditions that are codified in state laws. The Constitution does mandate that each state will have Congressional representation based on the size of its population, but it doesn't specify how the state determines its representatives. Presumably, a state could hold a state-wide election and give it's seats out proportionally. In small states with few reps, this wouldn't make much (or any) difference, but in larger states like California, with 53 House seats, a party getting 3% of the vote would be entitled to one seat.

What ideas do you have about solving the two-party problem?