You must keep in mind that the Balkans was a region in which all three empires had eyes on before the XIX century. Also, the Balkans, especially in the late XIX century became a playing field because there was the only place for these 3 empires the expand their influence and, in the case of the Turks, to keep their influence there.

I will start with the overall interests of the 3 states in the region. The Habsburgs fought the Ottomans since the 16th century in long and bloody wars that gradually pushed the front from the gates of Vienna towards the Balkans. The treaty of Passarowitz (1718) gave the Austrians parts of Serbia, Bosnia and Wallachia(Oltenia). This is the second great Austrian expansion after the treaty of Karlowitz in 1699. They lost almost everything that they won at Passarowitz after only 21 years. While the XVIII century saw many wars, the XIX century was quite peaceful between them. I will get to this in a minute. The important thing here is to remember the fact that Austria is no stranger to Balkan expansion.

Now for the Russians. The other great rival of the Ottomans was Russia. There were also many wars between them and more importantly, they continued in the XIX century. Putting aside the Crimean and Caucasus campaigns, the Balkan ones were very important. Russia, in the XIX century proclaimed itself as protector of the fellow Orthodox Christians. There was also a more... worldly prize: the Turkish straits. Controlling the straights would be of a huge importance from an economics POV and also military POV. Russia tried to expand towards the Balkans. In 1812 they took Bessarabia(today Rep. Moldova). After the war in 1829 they occupied the Danubian Principalities. After the 1877-78 war they occupied southern Bessarabia and they armies marched closed to Constantinople. This is where the other great powers stepped in and during the Berlin Congress pushed the Russians back a little bit. They got southern Bessarabia, but their way towards the Bosporus was blocked by the creation of independent Romania and the western support for the Ottomans that was keeping the "Sick man of Europe" alive just to stop Russia from expanding to the straights. Since the 1877-78 war and until WW1 there was no Russo-Turkish war because Russia was busy in other places. I will come back to this.

Now for the Ottomans. I will speak briefly because I mentioned them a lot when talking about the other two empires. They held the Balkans since the late middle ages. It was a spring board for further expansion(the way form Constantinople to Vienna is through the Balkans). For centuries, they were considered a great power by the rest of Europe because of their European possessions. The decline of the empire is best seen here, in the Balkans because in the XIX century they lost Greece, Serbia, Montenegro, Bosnia, Romania. In the years just before WW1 they lost almost all European possessions except a small part around Constantinople.

Now for the last decades of the XIX century and up until the Great War.

The Balkans was the only region in Europe where expansion was possible for the 3 powers.

After their defeat against the Franco-Italian forces in 1859 and against Prussia in 1866, the Balkans were the only region where Austrian interests could be advanced. It was a market where Austrian goods could have no match because the region had a weak industry. They had trade agreements with Serbia and Romania for example. In the case of Romania, it was signed in 1875 and reduced tariffs between the two. Austria dominated Romanian markets and the nascent Romanian manufacturing industries were severely affected because they could not compete with the modern Austrian industrial sector. Austria had tariffs wars with both Romania (in the 1880's) and Serbia(1906-08) and affected all the parties from a financial POV. Politically, Austria tried to expand their influence in the region by supporting a pro-Austrian dynasty in Serbia(Obrenovic dynasty) and an alliance with Romania, a country that became a member of the Triple Alliance since 1883. During the Balkan Wars, the Austrians and Germans became closer to Bulgaria than to Romania. Military, they occupied Bosnia and Herzegovina in 1877 and annexed it in 1908.

For Russia, after the win in the 1877-78 war, the region was of secondary importance for a few years. They had other things to do: like expanding in the East and south(Afghanistan, India, Iran). The Balkans again became of prime interest for them when the eastern expansion was checked by Japan and in the south, they came to an agreement with the Brits. Just before WW1, the Balkans was the only region where Russia could expand it's influence. They tried to attract other Slavic nations around here with their pan-Slavism, a kind of "brotherly" bond between nations of the same "race"(this was a term widely used in the era) and same religion. They had some success with this when Karađorđević dynasty in Serbia took power after a coup in 1903. They were pro-Russian and in this context you need to see the Pig war between Serbia and Austria in 1906-1908(it was a tariff war). They also tried to improve relations with Romania. It was important, because in order to enter the Balkans from Russia, you need to cross through Romania. In all Russo-Turkish wars of the XIX-century, the Danubian Principalities were either a theatre of war or Russian troops needed to cross in order to fight in Bulgaria, as it happened in 1877. Romania's army was also important in the region, as we can see in 1877 when Russia asked for help after some defeats in Bulgaria. Romanian intervention proved decisive. Anyway, there were talks for a dynastic marriage between the Romanian crown prince's son with one of the Czars daughter. It failed. But in 1914 there was a visit made by the Imperial family to Romania. After 1877 the Russo-Romanian relations were frosty(they took southern Bessarabia from Romania while they were allied) but just before WW1 they looked to be improving.

As for Turkey, they started to be a minor power here. They were in constant decline, but of course they tried to cling on to their European possessions. They were not successful. As I mentioned before, Greece became independent. Romania, Serbia and Montenegro were under Ottoman suzerainty until 1878 but the Turks really had no overall control in those countries. Bosnia was nominally under Turkish suzerainty until 1908, but in reality it was under Austrian occupation since 1878. The first Balkan war eliminated Turkey as a European power. This, coupled with the defeat in the Italo-Turkish war in 1911-12 made some Turkish leaders like Enver Pasha to dream of a Turkic Empire in the east. For all the XIX century they were still an important player here, but exactly before WW1, they were for all intents and purposes almost gone. They could not compete economically and military with more modern economies and armies of Russia and Austria.

As for the other major powers like France, Germany and UK. In the latter decades of the XIX century, they were more involved in the Scramble for Africa. There were way more riches and territory to be grabbed in Africa than in the tiny and messy Balkans.

Sources and further reading:

1. The Balkans. Nationalism, War and the Great powers- Misha Glenny
2. History of the Balkans. XVIII and XIX centuries- Barbara Jelavich
3. The Sleepwalkers- Christopher Clark(this is great for all the entanglements before WW1)