It is my understanding that, during the time when liberation theology was developed, it was a very radical deviation from mainstream Roman Catholic doctrine. It had close ties with Marxism and socialism, some of it's adherents advocated overthrowing their governments, and it subverted traditional norms in the Church.
Is this true? If so, how did the Catholic Church respond to a radical movement sympathetic to communism?
Oh man, I'm actually writing a paper on this right now! I hope I can help!
The short answer is that they didn't really respond well (for the record, I am not trying to indicate whether or not that was a good decision; I'm just really interested in the whole thing!). According to Valley, "there was an element of deliberate and willful misunderstanding of the basis of Liberation Theology" by a handful of the bigwigs in Rome. Most liberation theologians weren't actually supportive of violent revolution, but a lot of the way they phrased things made it pretty easy to represent the ideas as explicitly Marxist, whether or not they actually were.
Because of this perception of a connection between Latin American liberation theology and Marxism, the CIA backed the conservative folks in the Church to further alienate the progressive liberation theologians and the Vatican was so stoked to work with the Americans. The instability throughout Latin America at the time served to further polarize the left and right sides of the Church in the region.
Looking through the eyes of Gustavo Gutiérrez, who really coined and defined the term "liberation theology," the Church had an immense responsibility to not only help the poor, but to stand with them in social revolution. This different viewpoint has been described as "bottom-up" which was a dramatic shift from the tradition "top-down" view of the Vatican - that is, the Church traditionally kind of "starts" with the Pope and works it's way down to the laypeople where the Church in liberation theology "starts" with the oppressed and impoverished and constructs the Church from there. Obviously, that idea would be a hard sell to the people in power.
An interesting example of the Vatican's resistance to liberation theology can be seen in the role of Provincial Superior of the Society of Jesus (the Jesuits) in Argentina in the early 1970s. Father Ricardo O'Farrell was the Provincial Superior in the time after the Second Vatican Council which was pretty unfortunate for him; his time in the role saw significantly fewer Argentinians becoming priests (there were 25 vocations in 1961 and in 1973, there were only two - Jeffrey Klaiber speculated that this had much more to do with Vatican II than it did with anything Father O'Farrell could've done). In addition to this, Father O'Farrell was known to be fairly supportive of (or at least open to) the ideas of liberation theology. Because of this, the Vatican made the unprecedented move to support an unprecedented petition to remove O'Farrell from conservatives within the province. O'Farrell was removed from office two years before his "term" would traditionally expire - which I feel it's again necessary to emphasize that this was unprecedented.
He was replaced by Jorge Mario Bergoglio, who was much more along the conservative line and undid much of what O'Farrell would have considered progress; students would no longer study any of those new-fangled philosophers like Hegel or Kant. They would only study Aquinas and other "old Church Fathers" under Bergoglio. Studying liberation theology was actually forbidden according to Father Rafael Velasco, who studied under Bergoglio (I guess this might be a good time to mention that Bergoglio is now better known as Pope Francis for those that just know they've heard that name somewhere!).
That's not to say that Bergoglio didn't care about the poor; he just had a significantly different interpretation of the Gospel (which may have been influenced by the volatile environment within which he was thrust into power). He subscribed to Pope Paul VI's (the Pope at the time) exhortation Evangelii Nuntiandi that emphasized the importance of spreading the message of Christ and that this should be the primary concern of not only the Church, but every Christian.
It's also worth noting that many of the groups which had a significant role in developing liberation theology (like the Latin American Episcopal Conference [CELAM]) never officially endorsed liberation theology - arguably to avoid more conflict with the Vatican.
Also worth mentioning (as more of an interesting point than an answer to your question) is that Cardinal Joseph Ratzinger (the future Pope Emeritus Benedict XVI) wrote a few solid criticisms of liberation theology on behalf of the Church starting in the 1980s.
I hope I answered your question! I'm happy to elaborate if I was unclear on anything!
References/Sources (check out Valley's book if nothing else!):
Francis. “Laudato Si’ [Encyclical on Care for Our Common Home].” Vatican City: Vatican Press, 2015. http://w2.vatican.va/content/dam/francesco/pdf/encyclicals/documents/papa-francesco_20150524_enciclica-laudato-si_en.pdf.
Gutiérrez, Gustavo. A Theology of Liberation. 11th ed. New York: Orbis Books, 1984.
Novak, Michael. The Spirit of Democratic Capitalism. 2nd ed. Lanham: Madison Books, 1991.
Rowland, Christopher, ed. The Cambridge Companion to Liberation Theology. Cambridge: Cambridge University Press, 1999.
Vallely, Paul. Pope Francis: Untying the Knots: The Struggle for the Soul of Catholicism. 1st US ed. New York: Bloomsbury USA, 2015.