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.