Shavuot has no specific rituals because it is all about receiving God's Torah.
All biblical Jewish holidays but one are distinguished by specific mitzvot (commandments), that attend their celebration: Rosh Hashanah's shofar, Yom Kippur's fasting, Sukkot's booths and "four species," Passover's seder and matzah.
The one conspicuous exception is Shavuot. Although the standard, traditional prohibitions of labor that apply to the other holidays apply no less to Shavuot--and while special sacrifices were brought in Temple times on every Jewish holiday--there is no specific ritual or "objet d'mitzvah" associated with Shavuot.
There are, of course, foods traditionally eaten on the day--specifically dairy delectables like blintzes and cheesecake. And there is a widely-observed custom of spending the entire first night of Shavuot immersed in Torah readings and study. But still, there is no Shavuot equivalent to the shofar or the etrog (citron used on Sukkot) or the seder.
The early-19th-century Hasidic master Rabbi Levi Yitzchok of Berditchev suggested that perhaps the mitzvahlessness of Shavuot was why it is called throughout the Talmud "Atzeret"--which means "holding back" and refers to the prohibition on labor. The lack of particular Shavuot mitzvot, though, may reflect something sublime.
The Essence of Passivity
Shavuot, although characterized by the Torah only as an agricultural celebration, is identified by the Jewish religious tradition with the day on which the Torah was given to our ancestors at Mount Sinai.
That experience involved no particular action; it was, in a sense, the very essence of passivity, acceptance of G-d's Torah and His will. The revelation was initiated by G-d; all that our ancestors had to do--though it was a monumental choice indeed--was to receive, to submit to the Creator and embrace what He was bestowing on them.
Indeed, the midrash compares the revelation at Sinai to a wedding, with G-d the groom and His people the bride. (Many Jewish wedding customs even have their source in that idea: the canopy, sources note, recalls the mountain held, according to tradition, "over their heads"; the candles, the lightning; the breaking of the glass, the breaking of the tablets of the Law.)
And just as a marriage is legally effected in the Jewish tradition by the bride's simple choice to accept the wedding ring or other gift the groom offers, so did the Jewish people at Mount Sinai create its eternal bond with the Creator by accepting His gift of gifts to them.
That acceptance may well be the essential aspect of Shavuot. A positive, active mitzvah for the day--an action or observance--would by definition contradict the day's central theme of receptivity.