In Mark Mazower's The Balkans, he says that the mountains in the peninsula did not present a barrier to invasion. Is this true? Why?

I'll take a stab:

This is actually (mostly) accurate. The mountains in the northeast of the Balkans, the Carpathians and the Transylvanian Alps, are completely bypassed by the ease of movement one has through the Danube River valley and the Pannonian Plains. If you look at a map, The Danube River valley allowed easy passage from east-to-west, and the Pannonian Plains (also along the Danube river) are also devoid of many natural barriers in the west aside from the river itself. Situated on the Danube are Vienna and Budapest, who would make up the Austro-Hungarian Empire, and this was one of the major land-holding empires in the Balkans for a reason. East of Dragoman Pass, the Danube River valley, is completely open even as it stretches north, around the Black Sea. This would be where nomadic groups or Russia would easily enter the Balkans, and also a point of contention for the Ottoman Empire.

Basically, there are plenty of mountains in the Balkans that may have isolated what would later become national groups from one another. But in terms of imperial expansion, there are major passes and avenues of conquest that would ignore the hilly terrain. Most of these were valleys. Control over certain passes---for example, Belgrade is situated right on the border of the eastern Danube river valley and the southernmost extent of the Pannonian Plain---became key for an empire like the Ottomans. (Speaking generally).

The southwest is a slightly different matter. The Dinaric Alps and extremely hilly terrain along the Adriatic coast made a number of coastal city-states vulnerable to maritime conquest, such as that of Venice. However, a maritime power like Venice wouldn't go into the interior, and these coastal bits would end up isolated from the broader Balkan territories behind the mountains. This is the case with Dubrovnik. If you look at modern-day Croatia, its long, skinny Adriatic coast territory illustrates this fairly nicely. I'm not a Greece expert but the rapid changing-of-hands over certain Greek islands like Crete and the Dodecanese in the 18th-20th centuries also shows maritime empires would have an easy time nabbing islands (or just administrating them) or coastal strongholds. But venturing into the interior would not be so easy. In contrast, the overland empires of the Ottomans and Austria-Hungary were always more concerned with the major valleys and the Pannonian plains. This dynamic changed when Russia took a more active role---then, the Black Sea became extremely important for regional tensions.

A question about the relation of the Balkan people to the mountains is a big one, indeed. If you're ok with broad strokes, the mountains and hilly terrain were where resistance would obviously be most fierce against imperial rule, but also proved to be serious obstacles when the Balkan national-states tried to consolidate themselves in the 19th-20th centuries. Montenegro, for example, enjoyed virtual independence from the Ottoman Empire much earlier than any of its neighbors due to its paucity of good land and the extremely difficult task of sending administrators to a mountain country filled with armed bands. These armed bands were subject to a romantic interpretation as the Ottoman Empire began to severely deteriorate in the 18th-19th century. Basically, the Ottoman Empire was slipping out of centralized control for a number of reasons, including insubordinate janissaries and ayans (who were provincial governors of sorts). Some famous ayans that actively fought against the Ottoman Sultan were Muhammad Ali and Pasvanoglu. How does this relate to the Balkans? Well, the policing of the Balkan territory by the Ottoman Empire was slipping just as the centralized rule was. Lawlessness became rampant as janissaries and ayans took on a more predatory role. (In Wallachia and Transylvania, Greek Phanariots also drew severe ire from their subjects. )

For example, in 1791 Pasvanoglu, nominally an ayan under the control of the Ottoman Empire to administer Balkan territory, was in open rebellion against the Sultan, with his own janissaries as his army. To defend themselves, as had been custom for a while, Serbs in the contested territory formed armed bands to protect themselves. Selim III, Ottoman Sultan, actually legalized these armed bands and called on their help to suppress Pasvanoglu's disobedience. Which they did, until 1798 when Napoleon invaded Egypt. The Serbs, in this case, weren't fighting for independence, but rather just increased autonomy and a return to normalcy. In 1804, the armed bands reconvened to fight under Karadjordje Petrovic (a Serb) against a group of four janissary officers called the dahi, who were killing Serb nobles in an attempt to restore their control after the aborted attempt to oust them before.

The armed bands were invaluable in the ensuing struggle, and were mostly reliant on villages and towns isolated in the hillier terrain around Belgrade rather than the city itself. The mountains provided both natural cover and a preservation of cultural life that could be considered more 'traditional,' although I don't like that word. However, the fierce individualistic streak of these bands also contributed to Karadjordje, and many other fledgling Balkan govt's downfall. After the uprising against Ottoman rule, there was a period where Karadjordje needed to consolidate his authority. That didn't happen, due to competing loyalties among families granted new status from controlling the guerrilla fighters. Instead, after independence was quashed out of his control and he fled to exile in Austria, Karadjordje was killed by a political rival, Milos Obrenovic, who at that time was trying to appease the Ottomans.

If it's helpful to look at this in a more theoretical space, the mountains and hills tend not to be the wealthiest places in a country. They can't grow much food, a pastural way of life is generally lower quality, and large settlements don't normally get built in them. The first thing an empire will often do in an area like the Balkans is resettle its subjects from closer in its core (Constantinople) to the newly-conquered areas...and in that case, they better put them in the best spots. The valleys. Thus, the valleys tend to be the most fertile, rich areas for the new imperial cities. Those who resist imperial control or stick to a pre-imperial way of life are more likely to stay or flee to the mountains around them. This is the case with Kosovo, the part of the Balkans I'm most familiar with: much of the KLA that later instigated Kosovo's independence were what a typical American would probably call hillbillies---but that only makes sense.The attempt to exert colonial control over compounds in the Balkan hills, like that of Adem Jashari, provoked violent responses because these were the spaces where the most ethno-national traditions were enforced.

Hope that makes sense. To be frank, I'm not an expert on Greece or Romania, which from what I understand are a bit more...complicated, when it comes to the mountainous parts. Maybe someone else will come along and contribute.

Sources: History of the Balkans: 18th-19th Centuries, History of the Balkans: 20th Century, both by Barbara Jelavich.

The Establishment of the Balkan-National States, 1804-1920, by Charles and Barbara Jelavich.

Kosovo: A Short History by Noel Malcolm.