By chance several sources from Avalanche Press, and other works, have given me an idea that there may be a way for Austria to extend down to the Aegean Sea - if it should take Constantinople is strictly up for grabs.
The exerts are from 2-3 paragraph scenarios, so there are some generalizations for the players.
From Dreadnoughts p.18:
'During the 1780's, Austrian Kaiser Josef II formed what he terms his "Eastern Project," a plan to extend Austrian rule over all the European land held by Ottoman Turkey. He saw the great port of Salonika as the key to this; in Austrians hands, it would became the great marketplace where East and West did business. In 1912, some Austrian leaders urged occupation of Turkish Macedonia, including the huge port, lest it fall into unfriendly Bulgarian, Serbian or Greek hands. From there, not only could trade profits be hand but Austrian naval power could be projected eastwards."
***** I think the editors at AP have made a mistake and mixed up Josef II for his father Franz II.
from p.56:
"As early as 1815, Austrian naval officers pointed out how useful a 'well-ordered squadron' of fast warships would be to assert influence in the Mediterranean Sea...."
From The Great War at Sea: Mediterranean p.57:
"When the Congress of Vienna re-ordered Europe in 1815, the Austrian Empire received the old territories of the Republic of Venice. These included the island of Corfu and six others along the Greek coast. Unwilling to take such far-flung responsibility, the Austrians passed them to the British, who ruled the "Seven Islands" until 1864. "Had the Ionian Island of Corfu...belonged to Austria," Austrian naval historian Anthony Sokol wrote, "the Austrian fleet need not have been locked in the Adriatic; communications with Turkey would have been possible, and some of Austria's trade with the Levant would have been maintained."
Right now I can't really verify exactly what Josef II thought about the Eastern Mediterranean or the bit about Austria getting, but turning down, the Ionian Islands. However, I agree that a POD very early in the 19th century is needed for Austria to really do anything.
So here are some free floating ideas that at least may but Austria in striking distance at Constantople by the late 19th century.
1. Kaiser Franz, either by early death or by being very unpopular for siding with Napoleon too often, is forced to abdicate. So his removal from the scene could be anywhere between 1810 to 1815.
2. Archduke Charles becomes either Emperor or Regent for Kaiser Josef of Austria. Whatever way Charles and his brother Archduke John get into power and begin various reforms that get the country back on its feet. The Habsburg Court is certainly cleaned out.
3. Austria begins to industrialize and modernize throughout the 1830-1850s.
4. Political reforms eliminate most social unrest and increase loyalty to the Crown. There will be a slow evolution towards a federal type of government representing all nationalities.
5. The Hungarian revolt of 1848 is crushed by the Austrians and Hungary is broken into smaller states - like Transylvania, etc.
6. Corfu and the Ionian Islands are retained by the Austrians from the Congress of Vienna. Corfu becomes Austria's major Mediterranean port. Since 1815 it will be the policy of Austria to acquire a land route down towards the islands.
7. 1870s Prince Milan Obrenovich (1854-1901) of Serbia sells his country to Austria. The Habsburg Court also arranges the marriage of his son Alexander to a Habsburg princess to united the crowns.
8. 1878 Congress of Berlin - Austria acquired Bosnia-Hercegovina.
9. 1880-1890s Austrian annexation of Bosnia-Hercegovina and a 'short victorious war' that nets it Albania and Macedonia, along with Salonika. Granted this will be fighting it mountainous areas, but an enlarged Austrian fleet, along with help from the Greeks - who get territorial gains also.
10. Regarding Germany. In this ATL it is the Austrians that develop 'Drag noch Osten" (if I can remember my German correctly). The Austrians turn over their leadership in the German states to the Prussians. There is no Seven Weeks War.
11. No Italian unification. Tho the Kingdom of the Two Sicilies does acquire the Papal States.
Other odd ideas:
Napoleon II, the Duke of Reichstadt does live a long life, perhaps serves in the Austrian Army in the wars against the Turks.
Close ties between the Habsburgs and Romanov families. Alaska is given as a wedding gift from the Tsar of Russia to the Kaiser of Austria when he marries his daughter. (A very long shot) Perhaps a real place to be named "Franz Josef Land".
However, such a deal does get the Austrians to support the Spanish Empire, since one would need the Philippines to help get to Alaska. Which could lead to a Austro-Spanish-American War.
As a good sport, Prussia will annex Taiwan, as it thought about doing in the 1860s.
Just some ideas really.
The Ottomans halted all shipping through the Straits, thus stopping Russia’s profitable grain trade. During the next two years, the Russian Black Sea Fleet and a squadron operating in the Mediterranean blockaded the Straits, while France and Britain was helping out the Greeks (the French even landed troops in the Peloponnes in Octobet 1828). When the Treaty of Adrianople was signed in September 1829, Imperial troops were within 100km of Constantinople. 
