This is a complicated (but very interesting!) question. Anarchism understood as freedom from the state is modern in part because the concept of the state is modern, but, I think there are certainly a lot of thinkers and movements which shared a lot in common with Anarchism in a broad sense. You may be interested in Peter Marshall's book 'Demanding the Impossible: A History of Anarchism'. Its easily the best general history of Anarchism and spends a considerable amount of time on the pre-modern era. During the Middle Ages and the Early Modern period there were several radical millenarian movements and thinkers which shared a lot in common with Anarchist thought (in the very broad understanding of the term). Jochaim of Fiore, a 12th century monk who had an influential apocalyptic understanding of world history, argued that there were three ages to world history and that in the third and final age, which was approaching, all temporal authority would be destroyed and society would be ruled only by the teaching of the gospels. He had a considerable influence on a number of other millennial thinkers in the later medieval and early modern periods.

During the reformation, radical-democratic and communist ideas exerted considerable influence over a number of popular movements. Particularly interesting here is the German Peasants War, which was the largest popular uprising until the French Revolution, and was partly motivated by a radical-democratic Communistic vision. Thomas Muntzer, an ideological leader of the peasants, is said to have aimed at a society where all things would be held in common. Within the radical reformation, many of the groups and thinkers who advanced similar communistic and democratic ideals were termed "anabaptists" and they commanded considerable support. There are lots of studies on the radical reformation you might be interested in and Kautsky wrote a book on Communism in Central Europe at the Time of the Reformation which is quite good. Of course, the movements were suppressed and underground and it is often hard to reconstruct what political views the peasantry held. One really excellent book is The Cheese and the Worms by Carlo Ginzburg which explores the views of a radical Italian peasant who was condemned by the inquisition and uses it, in part, to explore peasant radicalism in the period which you would probably find interesting.

During the English Civil War there were many radical currents which shared some commonalities with Anarchism such as the Ranters (who were influenced by Jochaim of Fiore) and The Diggers. These are explored in a classic history by Christopher Hill, The World Turned Upside Down, which I would highly recommend.

Now your first followup question is very interesting. There were actually quite a lot of European religious reformers and radicals in the Early Modern period who actively took part in colonisation of the Americas and saw the colonial project as an opportunity to create their more perfect societies. An interesting example is Vasco de Quiroga, an key figure in early colonial Mexico, who was inspired by More's Utopia and attempted to recreate it in the colony. William Penn is an interesting example in North America. A related question is, to what extent did contact with the non-European world influence the development of European anarchism and political radicalism? In my opinion there is a lot of reason to suspect that it exerted a very significant influence. So take for instance Quiroga who attempted to create a society based on More's Utopia. But More's Utopia was itself inspired by the travel accounts of Amerigo Vespucci (the early explorer who America was named after). It was Amerigo's description of communist societies in the Caribbean which prompted More to imagine his more developed communist society in Utopia. In fact, when Europeans encountered America they came into contact with societies which shared much more in common with Anarchism than anything which would ever develop in Europe. Most native American societies had little concept of private property and were based on highly decentralised and democratic political models which guaranteed a substantial amount of personal freedom. It is impossible to prove, but I think it is quite likely that contact with these societies had a substantial influence on the development of radical thought in the early modern period and, ultimately, with the development of Anarchism. One historian who makes this argument is William Brandon in the book New Worlds for Old.

I am not really sure about your second question. I do not know much about ancient history and I'm not sure what pre-roman currents of Anarchism you mean. However, generally speaking classical thought was rediscovered starting in the 12th century when translations of lost Greek and Roman texts reached Europe through contact with the Islamic world. So classic works of pre-roman political philosophy, like Plato's Republic, reached Europe in this period and revolutionised European philosophy. In the early modern period there was a further rediscovery of Greek thought, led by humanist thinkers like More and Erasmus (so works like Utopia are also inspired by this rediscovery of Greek thought and, in fact, the name itself is a witty play on words for people who could read ancient greek). Herodtus's Histories (which talks about the Sythians who I suppose you can say had an anarchistic society) and Tacitus (who describes the German tribes in a positive way) were known in this period and exerted a considerable influence on European thought I believe. Also if you are referencing the Anarchist tendencies in Early Christianity they were of course very well known in Medieval and Early Modern Europe and exerted an extremely important influence on all radical thinkers (all of whom were of course Christian).