What grains are gluten free?
Note that wheat is a type of grass, and is technically a grain.

Grains without gluten

Not all grains have gluten&mdash;only those closely related to wheat do.  Grains which do not have gluten include:


Corn (maize) and its variants or derivatives such as cornmeal, polenta, hominy, or masa
Rice (all varieties)

A note on rice: some varieties are called "glutinous". This just sounds similar to "gluten", but is in fact used to indicate the rice's stickiness. Glutinous rice does not have gluten, not even as traces, its stickiness comes from a special type of starch. 
Millet
Sorghum
Oats (although many oat products are processed in plants that may have had wheat)


Sometimes confused for grains

Note that the following gluten free seeds are not grains, although sometimes they are confused for grains:


Wild rice
Quinoa
Buckwheat (despite the name, not a wheat nor a grain)
Amaranth


Non-grains used as gluten-free starches and flours

Therefore, all grain-like, starchy, or flour-type foods which are not grasses are always gluten free, including wild rice, quinoa (despite any other issues), and so on.

Additionally, flours or seeds based on non-grain seeds, nuts, and legumes are always gluten free, although there may be other dietary sensitivities.  These include:


Almonds
Soy beans
Peanuts
Chickpeas 
Mesquite flour
Amaranth
Quinoa
Wild rice


Finally, starches and flours based on roots, rhizomes, and so on are gluten free, including:


Potato
Sweet potato
(True) yams
Tapioca


Grains containing gluten

The following grains are all wheat relatives, and have gluten to a greater or lesser degree:


Wheat
Spelt
Rye
Tritical
Barley
Kamut


Note that the following are all processed wheat products or specific wheat varieties, and should be avoided to maintain gluten free diets:


Farina
Durum
Semolina
Couscous 




See also:  


Gluten Free Living
WebMD guide to gluten free flours