What did Eastern Han Chinese sound like?
The main go-to source is Schuessler's 2009 work Minimal Old Chinese and Later Han Chinese. 

As this "Later Han Chinese" is of the 1st century CE, it falls directly in between the Zhou Dynasty's Classical Chinese and Qieyun 切韵 of the Sui dynasty. It is the pronunciation of the 说文解字, of the 释名, as well as early Buddhist literature with all their transcriptions of Pali and Sanskrit.

However, Schuessler states that:


  The conservative strain of LHan, which is provided in this manual, could perhaps also be called Mid-Han; it can be converted to that of Eastern Han mainstream literature using the following paradigms: 


... and then proceeds to list six points, five of which are vowel changes, one of which is velar palatisation.

The relevant points under its general phonology section are:


No consonant clusters
Velar palatisation (notably of 支)
Tones emerge definitively
Final -s is preserved from Old Chinese
"Vowel bending" - a definite great vowel shift for Chinese in this period, intermediate between Old Chinese reconstructions and the system in Middle Chinese.