From what I've researched, more than Greek, the problem was Greek customs and religion that could threaten Roman traditions. Greek language and writings were just the vehicle for that influence to creep into the Republic. In his Natural History, Pliny quotes some words of Cato the Elder regarding the subject:

"Concerning those Greeks, son Marcus, I will speak to you more at length on the befitting occasion. I will show you the results of my own experience at Athens, and that, while it is a good plan to dip into their literature, it is not worth while to make a thorough acquaintance with it. They are a most iniquitous and intractable race, and you may take my word as the word of a prophet, when I tell you, that whenever that nation shall bestow its literature upon Rome it will mar everything; and that all the sooner, if it sends its physicians among us. They have conspired among themselves to murder all barbarians with their medicine; a profession which they exercise for lucre, in order that they may win our confidence, and dispatch us all the more easily. They are in the common habit, too, of calling us barbarians, and stigmatize us beyond all other nations, by giving us the abominable appellation of Opici. I forbid you to have anything to do with physicians."

A similar fear surfaced with the discovery of the alleged tomb of Numa, the second king of Rome, in 181 BC. There was no trace of the king's body, but two stone chests were found containing books in Latin and Greek. The Latin ones were said to be about pontifical law, whereas the Greek ones were said to be about philosophy; some even claimed it was Pythagorean (which would be very unlikely, given that Numa predated Pythagoras by two centuries). Regardless their authenticity, their contents were deemed dangerous, as explained by Livy:

"The books were first examined by the friends who were present. As the number of those who read them grew, and they became widely known, Q. Petilius, the City praetor, was anxious to read them and took them from Lucius. They were on very friendly terms; when Q. Petilius was quaestor he had given Lucius Petilius a place on the decury. After perusing the most important passages he perceived that most of them would lead to the break-up of the national religion. Lucius promised that he would throw the books into the fire, but before doing so said that he should like to find out, if allowed to do so, whether he could reclaim them either by the right of possession or by the authority of the tribunes of the plebs, without, however, disturbing his friendly relations with the praetor. The scrivener approached the tribunes, and the tribunes left the matter for the senate to deal with. The praetor stated that he was ready to declare on oath that the books ought not to be preserved. The senate held the praetor's asseveration to be sufficient, and that the books ought to be burnt as soon as possible in the comitium. Whatever sum the praetor and the majority of the tribunes thought a fair price for the books was to be paid to the owner. The scrivener refused to accept it. The books were burnt in the comitium in the sight of the people in a fire made by the victimarii."

It must be noted that just a few years earlier, the Bacchanalia (introduced in Rome due to Greek influence) had become a source of scandal and conspiracy theories. It was important enough for the Senate to decree the famous Senatus consultum de Bacchanalibus, and attitudes towards Greek customs in that era should take that atmosphere into account. Of course, Greek language and writings ended up overcoming those misgivings, to the point that, by the time of Claudius, the emperor himself would speak fondly of it:

"He gave no less attention to Greek studies, taking every occasion to declare his regard for that language and its superiority. To a foreigner who held forth both in Greek and in Latin he said: "Since you are already with both our tongues"; and in commending Achaia to the senators he declared that it was a province dear to him through the association of kindred studies; while he often replied to Greek envoys in the senate in a set speech."