We really have abundant false names and hypocritical formulae, especially in some formal and public speeches the truth of which is not the concern of the speakers. （self-made）
Many people feel uncomforatble with the use of the structure "natural object + whose +xx". So I will stick to use the incontroversial structure. But the way it is used here makes the whole sentence too long and awkward. How to write it in a more fluent way?
I agree with one of the comments that "the truth of which" is perhaps the least of things making this sentence awkward. It is difficult to know if a rephrasing loses any important nuance without more context around the sentence, but I might rephrase it as:
We have no lack of false names and hypocritical formulae, especially in public speeches, where truth is not the speakers' concern.
Some words that seemed superfluous: "really" and "formal" (can't public speeches typically be assumed to be formal?). Using "where" as a conjunction to replace the "of which" construction seems more natural to me than "whose", but I think "whose" would be acceptable too.
Perhaps the awkwardness arises from the butting of the two nouns "speeches" and "the truth".
The word order of the bolded phrase in the OP's sentence can be inverted
especially in some formal and public speeches of which the truth is not the concern of the speakers.
to relieve a degree of clumsiness.
You can also use:
. . . especially in some formal and public speeches where the truth is not the concern of the speakers.
