The equivalent of coffee beans would be loose leaf teas.
This is the traditional way to enjoy tea, and is generally preferred by tea connoisseurs. Just like whole-bean coffee, loose leaf tea keep fresh longer than bagged tea, and generally has a richer flavor. Tea bags are generally prepared by the cut-tear-curl (CTC) process, which breaks up the leaves more than orthodox tea rolling, meaning they steep faster and more consistently. As a rule, bagged teas are generally made from lower-grade leaves, and CTC processing helps hide defects. They may even be made from fannings; these are basically the dust and broken pieces that settle to the bottom of a pile of tea leaves. Both of these mean that bagged teas can be prepared more quickly, but the flavor suffers.
The tea leaves are prepared by measuring them out into a ball or basket, which is immersed in hot water to steep. Once the tea is steeped, the basket is removed to prevent the tea from developing a bitter, over-steeped flavor. There are also a variety of other ways to prepare loose teas.
Despite the generally better flavor, loose leaf tea is not as popular as bagged tea in some parts of the world, because you have to measure out the tea and use a utensil to prepare it. Tea bags represent a greater convenience, which explains their success.
Below are three examples of loose leaf tea. The differences in color and shape of the leaves are the result of different processes used in preparing the teas after harvest.
Loose Sencha (a Japanese green tea)
Loose Keemun (a classic Chinese black tea)
Silver Needle (one of the finest Chinese white teas)
Powdered tea is also available, although it is not as well known as bagged or loose tea. One of the best-known is matcha (shown below), a Japanese powdered green tea prepared by grinding up the whole tea leaf. It is generally prepared by whisking leaves and hot water together in a bowl until foamy, using a traditional bamboo whisk known as a chasen (pictured). The result is a powerfully flavored, heavier drink with a higher level of caffeine and antioxidants.
Powdered black teas are also popular with Indians, and are used in preparing masala chai. Spices are boiled in milk, and then powdered black tea is prepared in water separately. There are a lot of variations on this procedure, but it seems to be the most popular approach for Indian customers.
Note: Since I now work for a tea company, I could write quite a bit on any of these parts but am trying to keep the answer brief. If you feel like more detail would be helpful, post a comment and I'll expand on the subject a bit more.