It wasn't unconstitutional at all. Term limits were not added until 1947.

Edit: Why on earth does this comment have so many upvotes?

I don't think anyone in this thread has fully addressed OP's question. Whether or not it was constitutional (although OP never says it was or wasn't) is beside the point. Additionally, the fact that a healthy majority voted for FDR does not mean that there wasn't a sizeable minority of voters who didn't like the idea of a president having so many terms. I don't know nearly enough to try to answer that, but it is a good question regardless.

Were there people in America who opposed FDR's later elections to President on the grounds that doing so was fundamentally wrong? Was there a large body of people in America pointing at Cincinnatus and Washington and saying that FDR was going against the principles of Republicanism? If there were people saying this, what do we know about them?

Edit: /u/tayaravaknin has provided an answer.

I'm going to do my best to fully address the OP's question, because like /u/MrMarcusandSuperHead said, I don't think anyone really has.

Now it's already clear that FDR won a resounding victory in 1936, his re-election campaign for his second term. In 1940, he won a similarly resounding election, with over 84% of the electoral vote, and 54.7% of the popular vote. Now, in terms of why people voted for Willkie, and whether it had something to do with Roosevelt running for a third term, I've got a few sources on the subject.

In early October 1940, a Gallup poll asked respondents how they would vote if there were no war (WWII had begun, though the US had not entered). The vote showed a substantial change in voting pattern. Originally, the voter breakdown showed FDR with 49% of the vote, and Willkie with 43%, with 6% undecided. When asked the question of "What if there were no war in Europe?" (I'm paraphrasing here), they indicated a vote of 51% for Willkie, and 43% for FDR, and 6% undecided still. This indicates that the war was definitely a big shifter. But what about the 3rd and 4th term thing?

Well, in 1944 (for the 4th term election), 51% of the people who said they'd vote Dewey (according to a National Opinion Research Center [NORC] poll that was taken post-election, fairly soon after) because it was "Time for a Change". This poll, then, would suggest that people were likely tired of Roosevelt and wanted a new President. Does that mean it's because the third and fourth term? Not necessarily, but unfortunately I couldn't find any information on it specifically in polls...but it could be a variable in "wanting a change"!

It was certainly mentioned by the Republican party, in their party platform. The platform specifically mentioned that allowing a third term would "overthrow our American system of government", and they favored implementing an amendment to stop more than 2 terms. They included it again in 1944, though with less-harsh wording this time. As a side-note, other presidents did try to get third terms before FDR, but it had never been done (obviously), and never been so close as during the 1940 campaign. Whether or not that issue resonated with voters is the question, but I still can't find anything that discusses that. Hopefully someone else can!

Now, I do want to say one more thing: Willkie didn't stress the issue of a third term very hard in 1940. You can see that by viewing this book, which makes that point for me. There really wasn't much focus; even when Willkie talked about things like dictatorship, that was possible to interchange with things that the previous candidate opposing Roosevelt (Alf Landon) said along the lines of increasing government, rising administration power, lack of opposition, etc...it didn't necessarily have to do with the third term.

So did it have a huge effect? In most of the literature I've seen, no. I don't have the numbers to reinforce it, and while it may have contributed to some of the effects (probably moreso in 1944, though in 1940 I'm sure it could've factored in), it doesn't seem to have been anyone's main focus, anger, or anything like that.

Hope that helps, even a little :).

Other Source:

To Change or Not to Change Horses: The World War II Elections HELMUT NORPOTH Presidential Studies Quarterly, Vol. 42, No. 2 (June 2012), pp. 324-342

Followup question: what was public perception of FDR as he neared death? Nowadays people are wary of old presidents, fearing they'll go senile before their terms are over. FDR was also very sick late in life. Was he able to hide that from the public, like he did the polio?

I was told years ago in college that the Republican party of the 1980's made at least some sort of effort to repeal the 22nd amendment (apparently in the hope that Reagan would then run for a third term). The professor who told me this was a well respected scholar regarding the office of the presidency. Is there any truth to this?

After dominating the White House for almost a hundred years (only losing the presidency twice with Grover Cleveland and Woodrow Wilson), the Republicans took 5 straight swift beatings between 1928 and 1948. After the debacle that was the Hoover administration, which saw an intelligent but distant president mishandle the urgency of the Great Depression, the country was ripe for another, stronger candidate from the other side of the fence. Here you had a New York politician who pumped vigor into the office and gave the party back to the people. He utilized the radio to create the Fireside chats, putting him into the homes of millions of Americans with his calm, hopeful, and reassuring public addresses. Here was a man who finally understood the people, and was painted as the every-man. A true leader.

Roosevelt's circumstances are interesting because his elections often coincided with significant political events. For his first term it was the Great Depression. Before his second election, Americans decided to give the only man with a plan a second chance and elected him again. While the Depression was still was thing, it was winding down. It was during his third term that the United States entered World War II after the Japanese bombed Pearl Harbor. With America entering a war, it was seen as unwise to 'change horses midstream' and go with the Republican candidate. While he was a successful president in his own right, he owes a lot to some pretty substantial events in American history occurring during the administration.