The East is very sketchily mapped out in HoME 4 (The Shaping of Middle Earth), but whether or not the older cosmology present at that time remains valid post-LotR is another matter. The maps provided in that book certainly bear no relation to later concepts; there is very little to the east of the Blue Mountains, no Eriador, no Misty Mountains, no Mirkwood, no White Mountains, no Sea of Rhun, etc.
This, of course, is easily explained by the fact that the full geography of Western Middle Earth evolved during the writing of LotR (refer to HoME 7 (the Treason of Isengard) for the maps and discussion of this evolution).
What we do know is that further east there is (or was) the Inland Sea of Helcar, beyond which lie (or did lie) the Orocarni (Red Mountains, or Mountains of the East). However, as the Silmarillion states:
In the changes of the world the shapes of lands and of seas have been broken and remade; rivers have not kept their courses, neither have mountains remained steadfast...
So even if the old maps remain valid for the First Age, they shouldn't be considered valid for later Ages.
There is some further information in the Akallabeth; before the Downfall;
Thus it was that because of the Ban of the Valar the voyages of the Dunedain in those days went ever eastward and not westward, from the darkness of the North to the heats of the South, and beyond the South to the Nether Darkness; and they came even into the inner seas, and sailed about Middle-earth and glimpsed from their high prows the Gates of Morning in the East.
For Iluvatar cast back the Great Seas west of Middle-earth, and the Empty Lands east of it, and new lands and new seas were made...
And after it:
And those that sailed far came only to the new lands, and found them like to the old lands, and subject to death. And those that sailed furthest set but a girdle about the Earth and returned weary at last to the place of their beginning; and they said: 'All roads are now bent.'