This is not an answer but a longish comment, which moreover is certainly "subjective and argumentative". Reading all the stories given in the 12 answers, I find I can classify them in three categories:
(1) the stories that have no factual basis and are pure myths (e.g. the one about
Hilbert rejected by Gordan, or Grothendieck rejected by you-know-who, etc.). I'd like to add Fourier to this category but here I don't know the history well enough to be sure. What is certain is that Cauchy, faced with a contradiction between Fourier's result and a theorem he "proved" (limit of continuous function s is continuous) did not dismiss Fourier, and that others quickly dismissed (rightly) Cauchy's result.
(2) The ones that don't really concern mathematics: Boltzmann, Bolzano (whose work in mathematic become admired as soon as it was known, and was controversial for something else), Giordano Bruno, and even Brouwer, who as a mathematician was respected and even admired by about everybody else, and was only controversial as a philosopher of mathematics - and certainly no more than any other philosopher is controversial.
(3) The few ones that have a factual basis and do concern mathematics. I will restrict my attention to these cases as they are the only ones that really answer the question. Now,
I am afraid that in each of these stories, where a romantic genius makes a discovery that is ignored or rejected by the conservative establishment of mathematics, my heart is with
that so-called establishment, whom I can accuse of no wrongdoing, even with the benefit of the hindsight. Indeed, in none of these cases has the "romantic genius" been persecuted or even bullied (as was for example Giordanno Bruno, or to a lesser extent Gallileo). We, the mathematician community, have no auto-da-fé (not to speak of bonfire) in our history to apologize for. What happens
in all those cases is that there was a genial mathematician whose works suffered from serious shortcomings, and it was those shortcomings, and not the ones of the mathematical community, that made the process of assimilation of these works by the community
longer than it could have been.
Let me explain my point by discussing some cases:
Lévy, are we said, was a great probabilist, but not very rigorous. I agree with this description. Now is his work not being rigorous a plus, or a minus? To me the answer is obvious, and I hope everyone here agrees with that. At roughly the same time, Kolmogorov was founding rigorously abstract probabilities on Lebesgue's theory, and this gave his theorems
a convincing power that Lévy's have not. The mathematical community has done actually a
pretty good and relatively quick work in making Lévy's results rigorous and putting them in the mainstream theory.
Cantor is an interesting case. An absolute genius, for sure, with sometimes almost idiotic remarks -- like when he writes to Dedekind that his bijection between the line and plane refutes the basic idea of dimension. Dedekind kindly answers to him that people working in
geometry only consider continuous functions. Now it is perfectly normal and healthy that his works in set theory were exposed to such harsh criticism in his time. There were serious foundational problems in what he was doing. From the important point of view of rigor, he was putting mathematics back to the time of the early calculus, forgetting all the progress in rigor made in the nineteenth century, and indeed, there were as is now well-known some serious paradoxes hidden in his theory. The harsh criticism against Cantor's work (such as Poincaré's) was the anti-thesis in a dialectical process, where the role of the synthesis was played by lovers of the Cantor's paradise, that didn't want to lose rigor and admit paradoxes. Hilbert was forced by those very criticisms to develop a far-reaching program of mathematics in order to clear the discovered inconsistencies. Now the partial failure of Hilbert's program (Gödel's incompleteness and inconsistency theorems) shows that there really was something rotten in Cantor's paradise, and the indecidability of Cantor's favorite problem (the continuum hypothesis) retrospectively gives weight to Poincaré's criticism: arguably, Poincaré never asked a question which was later shown to be undecidable, unlike as with the continuum hypothesis or questions of the gender of angels.
Galois? Well, he has the best possible excuses for having written his genial discoveries
in such an unreadable way: he wrote them partly in jail, partly the night before his death,
and all before he was 22. Now for the very same reasons the mathematical establishment (the "Académie des Sciences", including people with a very different mind, like Fourier) have good excuses not to understand what he had done immediately. And again, very soon after his death (about 10 years after), his work was exhumed, intensely admired and integrated into living mathematics (especially by the German school).
PS: please feel free to vote down this unromantic post. My earlier self would probably have done so.