I agree with an earlier poster that they key to having multiple persons serve three consecutive terms would be to have Jefferson (or someone else early in the century) start the trend. Nevertheless, I wouldn't say that it's easy to get Jefferson to serve that long (he continually expressed a good deal of apathy and/or antipathy to his time in office). The most compelling reason I can think of would be a war in 1807, probably against Great Britain. If Jefferson does not serve three terms, it will be hard to get either Madison or Monroe to do so, both being more or less followers in that respect.
The next opportunity would have to be either Andrew Jackson or Henry Clay. The key to either would be to avoid OTL's election of 1824 (which can be done with some health butterflies to William Crawford, one way or the other). Note that I'd wouldn't expect both men to be able to do so in the same timeline. In either case, a war is probably needed to make the third term easily feasible in light of Washington's example, though if you fiddle with the dates, the 2nd BUS might be a reason for Jackson to stay on (and kill it).
Speculating on Lincoln first requires a description of what Reconstruction looks like under Lincoln, and even then I have a hard time seeing Lincoln have the energy or political capital to undertake a third term.
Probably the most plausible route, IMHO, would be to have the US more directly lose the War of 1812, though manage to keep Jackson a hero but have the British do something to provide a long-term thorn to the South (like maybe keep Florida Spanish, and or buy it themselves). The desire for revenge takes over as the South supports a more centralized government to enable a large US Army and Navy for a rematch sometime in the 1830s (in which case Jackson again becomes a nice candidate to lead such a rematch). The downside of such a change is that the cumulative effect will be to cause a significant divergence in the political culture (and probable borders) of the US, rather than a rather haphazard divergence of having a US that more-or-less resembles OTL's than happens by 1900 to have had a three term president.