Evans, J. The History and Practice of Ancient Astronomy, Oxford, 1998. An excellent source of material on Greek and Babylonian astronomy.
Krupp, E. C. Echoes of the Ancient Skies: The Astronomy of Lost Civilizations, Oxford, 1983. A popular overview of early thoughts about the cosmos and their relation to religion and time-keeping. Contains an extensive bibliography.
Thurston, H. Early Astronomy, Springer, 1994. An overview of early mathematics and astronomy, covering the cultures of interest to us. Moderately technical.
Maya
Freidel, D., Schele, L., and Parker, J. Maya Cosmos: Three Thousand Years on the Shaman's Path, Morrow, 1993. A recent account of Maya mythology and religion, showing how the myths persist to the present day. Relate the Maya myth of Creation to a map of the sky.
Milbrath, S. Star Gods of the Maya: Astronomy in Art, Folklore, and Calendars, Texas, 1999. Brings together a lot of material, but does not examine it very critically.
Sharer, R. J. The Ancient Maya, fifth edition, Stanford, 1994. A big book covering all aspects of Maya civilization.
Tedlock, D. Popol Vuh: The Definitive Edition of the Mayan Book of the Dawn of Life and the Glories of God and Kings, Simon and Schuster, 1985. This Maya myth, written down after the conquest, has an astronomical content.
Thompson, J. E. S. Maya Hieroglyphic Writing, Oklahoma, 1960. A comprehensive discussion of the first phase of the decipherment of Maya hieroglyphs. Covers most of the calendrical and astronomical information in detail.
Americas
Urton, G. At the Crossroads of the Earth and the Sky: An Andean Cosmology, Texas, 1981.
Babylonia
Neugebauer, O. History of Ancient Mathematical Astronomy, Springer, 1975. The basic reference on the decipherment of clay tablets and their interpretation. Probably more mathematical detail than you want to see.
Swerdlow, N. M. The Babylonian Theory of the Planets, Princeton, 1998. Many mathematical details, but has more discussion of the omen literature than Neugebauer.
Egypt
Lehner, M. The Complete Pyramids, Thames & Hudson, 1997. An excellent sourcebook on the pyramids.
Parker, R. A., Ancient Egyptian Astronomy, in Philosophical Transactions of the Royal Society, volume 276, 1974. A brief overview of basic information on Egyptian astronomy.
Antikythera Mechanism
Freeth, T. et al. Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism, 2006. Nature, Vol. 444, pp. 587-591. Note the detailed Supplementary info. Click here for the article.
Freeth, T. et al. Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism, 2008. Nature, Vol. 454, pp. 614-617. Note the detailed Supplementary info. Click here for the article.
Islamic (Arab)
Steele, J. M. A Brief Introduction to Astronomy in the Middle East, Saqi, 2008. A nice overview of Islamic astronomy. Includes material on Babylonian astronomy.
China
Clark, D. H. and Stephenson, F. R. The Historical Supernovae, Pergamon, 1977. A description of early observations of supernovae, the remnants of which are now observed by astronomers. Many of these early observations were made by Chinese astronomers.
Cullen, C. Astronomy and Mathematics in Ancient China: the Zhou bi suan jing, Cambridge, 1996. A translation of a treatise on astronomy and mathematics from the Han dynasty (1st century AD). The book also describes the developing mathematical and observational practices of ancient Chinese astronomers.
Needham, J. Science and Civilization in China, Vol. 3, Cambridge, 1959. This massive volume provides the most complete description of Chinese astronomy and mathematics in English.
Sivin, N. Cosmos and Computation in Early Chinese Mathematical Astronomy, E. J. Brill, 1969. A description of the cycles of time developed by the Chinese, as well as their early work on eclipse prediction. Quite technical.
Stephenson, F. R. and Green, D. A. Historical Supernovae and their Remnants Oxford, 2002. An updated version of the 1977 book by Clark and Stephenson.
Stonehenge and Related
Burl, A. The Stonehenge People, Dent, 1987. Not specifically on astronomy, but provides the archeological evidence on what we know about the people who built Stonehenge. Burl recognizes some of the astronomical alignments in Stonehenge, but regards it as much more a ritual site than an ``ancient observatory.''
Hawkins, G. S. Stonehenge Decoded, New York, 1965. A detailed astronomical interpretation of Stonehenge. Garnered a lot of attention. The archaeological community believes that he went too far (see Burl above).
Heggie, D. C. Megalithic Science: Ancient Mathematics and Astronomy in North-West Europe, Thames and Hudson, 1981. Uses statistical analysis to bear on the validity of claimed astronomical alignments (especially by the Thoms). Often skeptical, but does bolster some claims.
North, J. Stonehenge: A New Interpretation of Prehistoric Man and the Cosmos, The Free Press, 1996. A recent astronomical interpretation of Stonehenge and other megalithic structures. Many of the claimed alignmentments are related to stars. I did not find the arguments convincing.
Ruggles, C. Astronomy in Prehistoric Britain and Ireland, Yale University Press, 1999. In my opinion, this is the most sensible treatment of the subject. The result is that there is considerably less astronomy in prehistoric Britain and Ireland than has been claimed.
Thom, A. Megalithic Sites in Britain, Oxford, 1967; Megalithic Lunar Observatories, Oxford, 1971; (with A. S. Thom) Megalithic remains in Britain and Brittany, Oxford, 1978. The Thoms surveyed many megalithic sites. They probably overinterpreted their results (see Heggie above).
These are scholarly books. There are also many popular books related to archaeoastronomy that present an exciting story but present evidence only for their point of view. In this category are books on Egyptology by Graham Hancock and Robert Bauval.
JOURNAL
Also, the Journal for the History of Astronomy contains some articles on ancient astronomy.
WEB SITES
http://www.wam.umd.edu/\~\,tlaloc/archastro/cfaintro.html
Web page for the Center for Archaeoastronomy at the University of Maryland.
Exploratorium website of Chaco Canyon and Chichen Itza
These sites can be used as starting points to find many other related sites.
A great web site for eclipse information is
http://sunearth.gsfc.nasa.gov/eclipse/eclipse.html