China is quite a special case, since modern-day Han Chinese have a direct ethnic and cultural link stretching back into antiquity (notice, for instance, that we still call ourselves Han). A few specific ways in which Chinese have shown an interest in the past that parallel western interest in theirs are.

1. Poetry - Western classicism holds the poetry of Greek and Roman poets like Sappho and Ovid in high regard. In the same way, China has the poetry of the Tang Dynasty (and, to a lesser extent, the Song Dynasty), The most popular poets are Du Mu and Li Bai.
2. Literature - Just as the West has classic epics like the Iliad, Odyssey, and Aeneid, China has the "Four Classical Novels". These are Journey to the West, Romance of the Three Kingdoms, Water Margin, and Dream of the Red Chamber.
3. Visual Art - Just as Renaissance sculptors mimicked Greco-Roman sculpture, as in Michaelangelo's David, classical Chinese art forms, like paper cuts and ink paintings, are still popular today.

Echo pretty much everything zhemao said but for Iran.

Arab invaders destroyed a lot of Achamendied and Sassanid treasures, literature etc... But the Shahnameh is the epic that remains, it tells Iran's story through a mixture of history and mythology (its unclear where the line can be drawn) from the dawn of zoroastrianism, all the way up to the downfall of the Sassanids. This 50,000 verse epic is one of the main reasons that Persian is still spoken in Iran today and that Iranians have maintained a lot of their pre-islamic traditions in contrast to other places in the region that were arabised.

http://en.wikipedia.org/wiki/Shahnameh

It is hugely loved all around Iran, and has been for centuries.

Goethe puts it nicely on on the significance of linking to modern Iran. (Just as true now.)

Goethe was inspired by Persian literature, which moved him to write West-Eastern Divan. Goethe wrote:

When we turn our attention to a peaceful, civilized people, the Persians, we must—since it was actually their poetry that inspired this work—go back to the earliest period to be able to understand more recent times. It will always seem strange to the historians that no matter how many times a country has been conquered, subjugated and even destroyed by enemies, there is always a certain national core preserved in its character, and before you know it, there re-emerges a long-familiar native phenomenon. In this sense, it would be pleasant to learn about the most ancient Persians and quickly follow them up to the present day at an all the more free and steady pace.