I get the impression that the answer is indeed "We don't really know." This page summarizes a 1970s article by Beatrice Bodart, which I was able to find on JSTOR (free to read online if you sign up for an account):
Rikyu's execution was apparently rather surprising to a number of contemporaries, which suggests the actual reason was not at all well known at the time. There were speculations that it was for venality or corruption -- that Rikyu had been making money by selling fancy tea utensils (although there's no evidence of any great wealth when he died), or that a statue of him on a temple gate (placed there in gratitude for Rikyu's donation) had offended Hideyoshi because he had to walk under it when he visited the temple -- but there's no real evidence for any of that. Later, more romantic speculation had it that Rikyu objected to Hideyoshi wanting to take one of Rikyu's widowed daughters as a concubine; but apparently none of Rikyu's daughters were actually widowed when he died, so...
(The movie Sen no Rikyu suggests that Rikyu had criticized Hideyoshi's megalomaniacal plans for invading Korea and China, but this seems to be something invented by a historical novelist in the 1960s.)
Bodart basically argues that it was most likely the result of scheming by one or more of the younger, ambitious members of the government -- possibly Ishida Mitsunari -- who saw Rikyu as an obstacle to their influence. When Hideyoshi's powerful half-brother Hidenaga, who had long supported Rikyu, fell ill and died in early 1591, Rikyu became vulnerable. The outcome may have had something to do with an internal conflict over how to deal with Date Masamune, and perhaps with Hideyoshi's state of mind after the death of his half-brother and the illness of his newborn son, but it's unlikely we'll ever know.
[Edited to add: Just to round out the set of "theories" and answer the revised question a bit better: There's also an argument that he was executed for converting to Christianity; Bodart thinks this is very unlikely, given his close relations with Buddhist priests, his donations to Buddhist temples, and his Buddhist-inspired death poem. Other theories are that he tried to poison Hideyoshi, that Hideyoshi and Rikyu differed over the aesthetics of tea ceremonies, and that he was an affront to the strict social hierarchy Hideyoshi was promoting because of his origins as a merchant's son. Bodart doesn't think much of these, either.]