Since acts of worship in Hellenism, like those in other modern pagan religions, are usually called "rituals" in English, it will be helpful to define just what we mean by that word in a religious context:
A ritual is a complex of actions effected by, or in the name of, an individual or a community. These actions serve to organize space and time, to define relations between men and the gods, and to set in their proper place the different categories of mankind and the links which bind them together. 
The main type of ritual in Hellenism is the sacrifice, in which material things are offered to the gods to assure their goodwill and blessings, or to thank them for the gifts they have already given us. From the earliest phases of Hellenic religion, "the bond between man and the sacred is consummated in the continuous exchange of gift for gift."  Just as human social bonds are reaffirmed by gift-giving, the voluntary aid of the gods is sought through sacrifice.
Jan Bremmer writes:
...we reach a better understanding when we consider the relationships between gods and mortals as analogous to those between princes and commoners. Although gods did uphold the rules of justice, their obligations to kin and friends had priority. 
The goal of sacrifice, then, is to establish the bonds of friendship between gods and humans, and so encourage divine goodwill.
How do Hellenes imagine the gods responding to their offerings? Since the gods are in a wholly different order of being from mortals, it is apparent that their needs are different from ours. Homer tells us that the gods enjoy ambrosia and nectar, not human food and drink. Indeed, human blood does not run in their veins, but a substance called ikhor. We may further imagine that even this anthropomorphism, which scholars tell us is so characteristic of Hellenic religion, is one of the greatest "true lies" of myth: Few moderns truly believe that Zeus is sitting in the clouds taking deep, appreciative whiffs of the roasted barley on our sacrificial fires, and many thoughtful ancient people arrived at the same conclusion.
Looking at what we can surmise of divine nature, we can say that
...the sense in which men need the gods is quite different from the sense in which the gods need men. Men live by the hope of reciprocal favour, charis. 'It is good to give fitting gifts to the immortals'--they will show their gratitude. But it is never possible to count on this with certainty. The ritual, it is true, is attended by the expectation that it will produce certain effects, but the Homeric gods can always say no without giving any reason. 
Does this mean that ritual is a waste of time, an exercise in futility? Hardly. The human spirit needs a point of focus in order to fulfill its potential, and religious ritual provides just such a focal point. In ritual we express, with potent symbols, our understanding of our place in the world and how that place relates to the state of being we call "divine." We give voice to feelings of eusebeia, piety: We display our gratitude, our hopes, our fears, our joy in life, all in a context that sanctifies our human experience. This we do as a community, a community of mortals:
This is not an exchange of gifts celebrated by a hierarchical society of gods, priests, and commoners: together at the same level, men and women stand here about the altar, experience and bring death [in animal sacrifice], honour the immortals, and in eating affirm life in its conditionality: it is the solidarity of mortals in the face of the immortals. 
Types of Sacrifice
The centerpiece of Hellenic worship is an offering of food—formerly animals, now primarily grain, fruit, and other bloodless foodstuffs—and its common counterpart, libation (drink-offerings or spondai). The offering of food, some of which is normally burned, is called thusia, and is the typical form of sacrifice to the Olympians.
A sacrifice can be as simple as a little barley meal or incense prayerfully cast into the fire  and a few drops of wine poured out onto the ground, or as elaborate as the hecatombs (hundreds) of cattle that the ancients offered on special feast days. (Of course, today we would substitute wheat cakes or loaves of bread for the cows!) Since most modern Hellenic ritual is performed on a small scale, our sacrifices will consist of common foodstuffs—bread, fruit, wine, milk, honey, olive oil—and incense. We may also dedicate our own handicrafts to sacred use. This is in keeping with the fundamental practice of "first-fruit" offerings: "The pious man takes to a sanctuary a little of everything which the seasons bring, seasonal gifts (horaia), ears of corn or bread, figs and olives, grapes, wine, and milk." 
Whatever is offered should be of the highest quality possible given our circumstances: This is our gift to the gods and expresses our esteem for them. It should be, in fact, a sacrifice: a giving of your life energy, whether in the form of time, effort, or money. Although a few organic apples might cost the same as a big bag of potato chips, the former is an appropriate offering, and the latter, most people would agree, is not. Just as you make special efforts to please guests with a pretty table with flowers and your best linens, tasty food and drink, and a comfortable, inviting atmosphere, so in sacrifice we show our hospitality to the gods, our honored guests.
In addition to festival sacrifices which occur on a regular basis and on behalf of the whole community, individuals may make private offerings to the gods. One common type is a votive offering, one made as the result of a previous vow.  Who has not prayed in desperation, "If you get me out of this mess, I promise I'll pay you back!" For Hellenes, such a prayer is an oath and places a sacred obligation on the one who so swears. For this reason we find monuments to military victories that consist of votive offerings of weapons—war making prayers in desperation a daily necessity—and healing sanctuaries full of thank-offerings from successfully healed patients. Today, we may promise a special ritual, a new home-shrine, a change of lifestyle, or many other things should our prayers be answered.
So far we have been discussing sacrifices made to the Olympian gods. But there is another type of sacrifice, that offered to chthonic (earthly or underworld) deities, to the heroes, and to the dead. There are a number of signficant differences between the two types of sacrifice, although it should be borne in mind that usage varies enough that these distinctions cannot be considered absolute. The primary differences are as follows:
- When sacrificing to the Olympians one speaks of thusia; sacrifice to the Khthonioi is called enagisma.
- In animal sacrifice, the animal's throat faced upward for the Olympians and downward for the Khthonioi, so that the blood would flow directly into the ground.
- Altars to the Olympians, called bomoi, are set up from the ground; offerings to the Khthonioi are placed on a low altar called an eskhara or into a trench (bothros) dug in the earth.
- Animals offered to the Olympians--usually cows and oxen--were white; the Khthonioi received black animals associated with the earth, such as rams and pigs. Bloodless offerings of crops and especially honey are common to the Khthonioi.
- A drink-offering to the Khthonioi is called a khoe; the entire contents of the pitcher is spilled out onto the earth.
- The Olympians were honored at temples, often set on high ground; the Khthonioi in caves or underground sites.
- Olympian sacrifice typically happens in the morning; Khthonic in the evening or night.
- When praying to the Olympians, one raises one's hand upward in supplication; to the Khthonioi, one faces the palms down toward the earth. 
Other Types of Worship
Ancient Hellenic worship consisted of more than just straightforward sacrifice. In fact, one of its unusual features were the agones, or contests, held in honor of the gods. These might be athletic contests, such as the famous Olympic Games, or dramatic, poetic, or musical contests. It may be difficult for modern people to imagine a sports event as a religious service—although we certainly bring enough fervor to them today!—but this notion was intrinsic to the Hellenic idea of worship.
Today, sadly, our numbers are still too few to revive the agones on a grand scale, but we can integrate the idea into our worship in small ways. Instead of writing and acting full-scale dramas, we can memorize and present a single speech from one of the great dramatists at the City Dionysia. We can tell jokes instead of acting in comedies.  In place of extensive torch races, young people can run a short course with a lit candle. Where there are not enough children for a chorus, individuals can sing solo or in small ensembles. Poets may offer their best efforts to the gods. In Hellenism, personal achievement, when dedicated to the glory of the Immortals, is a cause for celebration, and a celebration in itself.
Notes Bruit/Schmitt, p. 27.
 Burkert, p. 35. (Gendered language as in original.)
 Bremmer, p. 11.
 Burkert, p. 189. (Gendered language as in original.)
 Burkert, p. 53.
 Burkert, p. 62: "to strew a granule of frankincense in the flames is the most widespread, simplest, and also the cheapest act of offering."
 Burkert, p. 67. (Gendered language as in original.)
 Burkert, p. 68.
 Guthrie, pp. 221-222. Guthrie's entire chapter on the Khthonioi is well worth reading.
 My thanks to Pyrokanthos of Thiasos Olympikos for this idea.
By Drew Campbell, "Old stones, new temples", Chapter 8.