From Answers.com (Start at the easiest to find, and work down.):

VEG IN THE ANCIENT WORLD

(Greek physician Diocles of Carystus)
beet greens, mallows, sorrel, nettle, orach, iris corms, truffles, and mushrooms

(archeological remains in the Fertile Crescent)
cucumbers, chards, gourds, onions, garlics, leeks, melons, chickpeas, lentils, cress,
kales, and sesame—both for the seeds and for the oil. Colocynth melons,
which resemble small watermelons, were grown primarily for medical
applications.

(Phoenician traders introduced to N. Africa and S. Spain)
shallots, artichokes, saffron

(Egyptian papyri and tomb paintings)

The most commonly mentioned vegetables were lentils, leeks, lotus, melons, gourds, garlic, asphodel (grown for its bulb), fava beans, chickpeas, fenugreek (ground as flour), garland chrysanthemum (now popular in Asian cooking), cucumbers, onions, lettuce, and mallow.
Egypt also served as a conduit for the introduction of watermelons from
tropical Africa and, during the late Ptolemaic Period, for the
introduction of rice, taro, and sugar cane from Trapobane (ancient Sri Lanka).

The Greek occupation of Egypt under the Ptolemys radically altered the
Egyptian vegetable garden, both with new introductions and in lasting
terminologies. Molókhe or mallow (Malva parviflora),
which was once so important to Greek cookery both as a green sauce and
as an ingredient in complex recipes, also supplied leaves used like
grape leaves for making dolmas. The Egyptians transferred the Greek name to a native wild plant now known as Jews mallow (Corchorus olitorus), which was similarly used in green sauces. It is still called molkhia
in Egyptian Arabic. The use of the same name for plants of a different
genus or species is one of the lasting ambiguities inherited from the
ancients, who were more apt to lump vegetables together according to
how they were used, as in the case of the Roman propensity for treating
carrots, parsnips, and parsley root as pastinaca.

Apparently I need a copy of Apicus.

Cabbage, especially the kales, originated in northern Europe…This is evidence in itself that the vegetable exchange during Roman
times was complex and two-way, with the Romans themselves learning new
things from conquered peoples.

The most commonly mentioned vegetables in Roman literary sources
include many still known, although in shape and habit they probably did
not resemble modern varieties. The list includes turnips, radishes,
rocket (arugula to American grocers), leeks, lentils, lettuce, orach, Old World gourds (eaten young like zucchinis),
cabbage, onion, peas, chickpeas, fava beans, cucumbers, asparagus,
cowpeas, beets, beet chards, sprouting broccoli, watermelon, garlic,
mallow, dock, chickling vetch, and blite—otherwise known as purple amaranth (Amaranthus lividus)…barley, wheat, lettuce, and fennel…cyprian lettuce.

Well, that’s a start.  I’ll condense the Arab imports and Renaissance tomorrow, and sort out some planting lists.

Here’s an answer to my sunlight problem:

Furthermore, it is clear from most of the records, whether Assyrian,
Babylonian, or of any of the other cultures sharing the Fertile
Crescent, that vegetables were commonly grown around date palms or
fruit trees. The palms filtered the blasting sunlight and gave the
garden the appearance of a welcoming grove.