Author's note: This post was partially updated 12/20/17.
Figuring out what the ancient Maya ate is till an ongoing process. Currently, it is understood that the Maya obtained food from both animal sources as well as plant sources (some of which it is thought were developed during the Archaic Period). Also understood is that different regions that the Maya civilization lived in had different resources generally available, which affected the everyday diet of the local communities.
Examples of the produce that the ancient Maya grew and then ate include -- but is not limited to -- cacao fruit (Theobroma cacao,) sweet potato (Ipomoea batatas,) chilies (Capsicum annuum,) corn or maize (Zea mays,) squash (Cucurbita pepo,) guava (Psidium guajava,) manioc (Manihot esculenta,) sapodilla fruit, chaya (Jatropha aconitifolia,) manioc/cassava (Manihot esculenta) the common bean (Phaseolus vulgaris,) avocado (Persea americana,) and vanilla (Vanilla planifolia.)
The ancient Maya processed their corn by boiling it with snail shells or with white lime. The process, known today as nixtamalization, made the corn's niacin available. Kinds of foods made with corn include tamales, corn beer, atole (a breakfast drink, according to bishop Landa) and the ancient Maya style of tortillas (thicker than the Aztec version) -- to which chili peppers, honey, achiote or squash seeds that had been toasted and ground were added.
Tamales were a often eaten lunch item. They could be filled with meat fillings, iguana eggs, flowers (for example, squash flowers), green vegetables and toasted squash seeds. Wrapped in leaves, they were cooked various ways such as under coals, steamed inside a certain kind of jar. Where it was used, the Maya would use a comal to cook tamales.
And how did people drink their atole? Those who are understood to have been commoners put honey, chili peppers, squash seed powder or herbs in their atole. On the other end of the spectrum, people understood to have been elites would mix the fermented, roasted and ground up seeds of the cacao tree into their atole.
Examples of animals that the ancient Maya hunted that you may of heard of included manatees (where there were communities on the coast,) iguanas, foxes, rabbits, white-tailed deer, possums, anteaters, and fish. Some animals that you may not of heard of included brocket deer, agoutis, coatis, kinkajous, tapirs, pacas, and peccaries.
But they might not have just hunted some of these animals. The ancient Maya may have kept some of them, such as peccaries and ocellated turkeys.
The ancient Maya also had several domesticated species. One of these was the turkey. The other was a certain kind of dog.
Grilling -- via skewering the meat and putting it on a wood frame over a fire -- was common for dog meat, deer meat, bird meat and peccary meat, and may have been common for iguana and turtle meat. Roasting food in a pit like the Hawaiians was also common: the meat was placed in a fire pit in the ground on top of hot stones, and the pit was covered. This process was common for festivals.
Though grilling was common, the most often used method to cook both fish and bird meat was to boil it. It is possible that boiled fish and poultry were used in stews - like tamales, a common lunch food.
Consideration: Dietary Theories
In terms of reconstructing which foods were most important, one interpretation was that corn was the most important to the ancient Maya. According to The Ancient Maya: New Perspectives corn may not have been as important as has been thought.
There is also the theory that corn, squash, and beans (the "three sisters" in certain North American cultures) as well as chilies were commonly eaten in both the highlands and lowlands of the ancient Maya world, with tropical fruit like the cacao tree's fruit were more common in the Yucatán Peninsula and in the Petén region.
As to how common meat was, that depended on the person's rank in society. Meat was not as common for people understood to be commoners to eat. Festivals were the time during which they would be more likely to eat it.
Google Books: "Encyclopedia of the Ancient Maya"; Walter R.T. Witschey; 2016
Google Books: "Daily Life in Maya Civilization"; Robert J. Sharer; 2009
Google Books: "Handbook To Life In The Ancient Maya World"; Lynn V. Foster; 2005
Google Books: "The Ancient Maya: New Perspectives"; Heather Irene McKillop; 2004
Google Books: "Animals & Plants of the Ancient Maya: A Guide"; Victoria Schlesinger; 2001
Google Books: "Archaeology of Ancient Mexico and Central America: An Encyclopedia"; Susan Toby Evans, David L. Webster; 2001
The Free Dictionary: Cassava