Many of the first large-scale, agricultural monasteries were in Ireland and Scotland. The life of Collum Cille, known to English speakers as "St. Columba", is one of the first records of monastic life. It was Irish clerics, not Romans, who developed the monastic system which was based on ascetic hard work and scientific husbandry.
It is true that many old Roman estates did get turned over to monks, but this was only well after the system had been developed and proven in Ireland. At the time, Roman methods were relatively inefficient, especially with regards to husbandry technique, such as sheep rearing, cattle rearing and bee keeping. The science of the monks, their hard work, and their scrupulous honesty, made estates run by Irish clerics immensely profitable. Because of this, the old Romans and barbarians like the Franks and Visigoths, literally fell over themselves trying to persuade the monks to come and take over their lands and make them productive and fruitful.
The reason why Frankish monasteries were often started on forested land is because it was less valuable than the agricultural land, so it was much easier to give it away to foreigners. When the monks received such lands they would gradually clear them--expensive and back-breaking work--turning them into fertile farmland. The local landlords and chiefs, who just wanted fast money, had no stomach for such hard work, so they left it for the monks.
As you rightly suspect, hunting grounds do not produce much income, but given time and hard work the monks turned those lands to the many of the productive fields you can find in France today. If you look at the village of Cluny today, where the abbey is, you can see the whole town is surrounded by agricultural fields.