"First published in 1973 - and followed by Volume II in 1976 and Volume III in 1980 - this anthology has assumed classic status in the field of Egyptology and portrays the remarkable evolution of the literary forms of one of the world's earliest civilizations. Volume I outlines the early and gradual evolution of Egyptian literary genres, including biographical and historical inscriptions carved on stone, the various classes of literary works written with pen on papyrus, and the mortuary literature that focuses on life after death. Introduced with a new foreword by Antonio Loprieno. Volume II shows the culmination of these literary genres within the single period known as the New Kingdom (1550-1080 B.C.). With a new foreword by Hans-W. Fischer-Elfert. Volume III spans the last millennium of Pharaonic civilization, from the tenth century B.C. to the beginning of the Christian era. With a new foreword by Joseph G. Manning" -- Amazon.com.
https://en.wikipedia.org/wiki/Miriam_Lichtheim
I've got, and have read, the three-volume edition of Ancient Egyptian Literature.
Almost everything before the Greeks, as far as Western and near-East literature, reads alien as hell and is pretty hard to get a grasp on, largely (I think) because so much of it exists absent context, with the remarkable and towering exception of Gilgamesh, which is truly amazing. I've tried Sumerian and Akkadian collections in translation, and bounced the fuck off after putting in substantial effort. Got zero out of it. It's all fragments of poems to gods that you have to go read a bunch of modern writing to figure out (where'd they get their info? It's not in the surviving literature, I have no idea). You need several times as much commentary as original text, to make sense of it.
Ancient Egyptian literature is the only of that set (Gilgamesh aside—everyone should read Gilgamesh) that I'd tentatively recommend to a general reader as the easiest to get into, though the entirety of Lichtheim's collection might be a bit much. Of interest are examples of the various genres of Ancient Egyptian literature—"mirrors for princes" instructional texts; scribal copy-practice instructional texts (why not impart some wisdom and/or entertainment with copy practice?); formulaic but evolving funerary inscriptions (they routinely threaten their gods, it's wild); monumental "historical" inscriptions (propaganda, LOL); and the handful of fictional tales, which tend to be entertaining enough but are in multiple cases frustratingly incomplete (talk about cliffhangers! You'll never get the ending, probably). It's a rough read, not exactly relaxing cotton-candy airport-novel reading, but far more tractable than the surviving fragments of the Fertile Crescent ancients (except Gilgamesh! Read that!)
Actually, there's another category of literature from that region, the written form of which isn't so old as those, but that has oral lineage with roots back into that era: the biblical "Old Testament", or Jewish scripture. It comes off rather well by comparison to the rest, though benefits from very-late composition and editing (contemporary with the Greeks, who read just fine to moderns) compared to the others. That's a fun one, too.
I read all three volumes of Lichtheim's Ancient Egyptian Literature first, though, so that may have spoiled me. It's dry as hell but way better than either of those other volumes. Was good enough that I remember quite a few details and individual works from it years later, unlike those.
I've got about seven volumes of ultra-early literature (the last 1.5 or so of Lichtheim aren't that early), six of which I've read or attempted (haven't made it to the Indian philosophy sourcebook yet, some of which is quite old) and of them the only one I'd recommend to the general reader is Stephen Mitchell's Gilgamesh, which was entirely excellent and so wildly better than any of the other Mesopotamian work from that time that I suspect Mitchell embellished heavily (IIRC reviews indicate he didn't, but still) or the modern version of the work was much improved by the time any of the copies or large fragments we have were written/chiseled. It's crazy good. A couple later Egyptian tales were of similar (though definitely lesser) quality but absolutely nothing else Sumerian or Akkadian's even got a hint of anything like that, from what I've seen.
I'm not sure I'd recommend the Sumerian or Akkadian collections to anyone, really, unless that's their specialty in which case they don't need me to recommend things to them. It's possible I just chose poor collections, though. Certainly the Akkadian one was no good, though I'm not sure there's any better in English.
1) The three volumes of Lichtheim's Ancient Egyptian Literature
2) Stephen Mitchell's Gilgamesh
3) Black et al, The Literature of Ancient Sumer
4) Foster's From Distant Days (Akkadian)
After that there should be some major works from India, but I haven't gotten around to that yet, so it's, uh, (counts) 14 volumes of Greek lit, philosophy, and math instead. A few Chinese authors ought to be represented along with the Greeks, but again, haven't gotten to it.
I usual prefer accuracy over readability in translations if I have to choose between the two, but I make an exception for some epic verse. Mitchell's Gilgamesh is one of those cases. It's not an abridgment or a retelling, but it errs on the side of readability. Great read, I'd definitely recommend it.
Lichtheim's introductions leave me wanting more, but those books are still very good, and they make the Sumerian and Akkadian volumes seem really bad in comparison. AFAIK there's not much out there better than those two, so those literatures may be in need of a good, well-edited anthology. The Literature of Ancient Sumer is serviceable, but I wasn't even able to finish From Distant Days and I'm considering getting rid of it. Part of that's the literature itself (seems like a big step down from Sumer, IMO) but part of it's the poor work of the editor, annotator, and translator, Foster.
Only the first few works in Vol. 1 of Lichtheim predate the Sumerian and Akkadian literature, and most of the stuff in Vol. 3 is all the way up in the first millennium BCE, but I keep the trilogy together anyway.
(1) http://www.amazon.com/Ancient-Egyptian-Literature-Middle-Kin...
(2) http://www.amazon.com/Gilgamesh-English-Version-Stephen-Mitc...
(3) http://www.amazon.com/Literature-Ancient-Sumer-Jeremy-Black/...
(4) http://www.amazon.com/Distant-Days-Poetry-Ancient-Mesopotami...
[EDIT] formatting [EDIT 2] If you're only going to buy/read one volume of Lichtheim I'd recommend Vol. 2 (not the one I linked)
There are a couple incomplete tales in that 3-volume work and I really wish they'd had a more stable storage medium, because now I'm stuck with accidental millennia-old cliffhangers.
LOTR, though, was too boring for me, just couldn’t keep going. I couldn’t stomach one more elf song or backstory of a tree. Though I acknowledge that’s probably some very-specific pathology of mine, rather than the book’s fault. I do seem to be wrong, in a sense, in this judgement.
Oddly enough, I later read The Hobbit, and loved that.
I do enjoy the Peter Jackson films (extended, especially for ROTK, which kinda sucks in its theatrical cut)
Almost all of that "genre" is of the "seen one, seem them all" variety. Most of it's very formulaic—think nicene creed or traditional church hymns, but less interesting. There were a bunch of standard texts that were reproduced on many tombs, and as time went on this body of necessary texts grew, so that's very likely what a lot of this is. Sometimes you'd get some passages of history but those are usually about as dry as the tombs themselves. More often those are on monuments, not buried in tombs, anyway.
"Mirrors for princes" (instruction texts for rulers, often both moral and practical in nature), scribal training texts (yes, really), and their few actual stories that survive, are all way more interesting than the stuff inscribed in tombs, in general. A lot of those don't survive in hieroglyphs, though, but in hieratic or demotic script.