not all civilizations built temples. in fact cults originated before temples, like animism.
I have in mind that, there is not only one way to offer worship.
What most closely resembles the ancient Roman cult is Roman Catholicism and its syncretism.
Sometimes the places of worship have to do with the descent or sighting of a god or goddess, or simply some natural phenomenon.
The most ancient and the most prominent asclepeion (or healing temple) according to the geographer of the 1st century BC, Strabo, was situated in Trikala. The 1st century AD Pool of Bethesda, described in the Gospel of John, chapter 5, was found by archaeologists in 1964 to be part of an asclepeion. One of the most famous temples of Asclepius was at Epidaurus in north-eastern Peloponnese, dated to the fourth century BC. Another famous asclepeion was built approximately a century later on the island of Kos, where Hippocrates, the legendary "father of medicine", may have begun his career. Other asclepieia were situated in Gortys (in Arcadia), and Pergamum in Asia.
From the fifth century BC onwards, the cult of Asclepius grew very popular and pilgrims flocked to his healing temples (Asclepieia) to be cured of their ills. Ritual purification would be followed by offerings or sacrifices to the god (according to means), and the supplicant would then spend the night in the holiest part of the sanctuary– the abaton (or adyton). Any dreams or visions would be reported to a priest who would prescribe the appropriate therapy by a process of interpretation. Some healing temples also used sacred dogs to lick the wounds of sick petitioners. In honor of Asclepius, a particular type of non-venomous snake was often used in healing rituals, and these snakes— the Aesculapian Snakes— slithered around freely on the floor in dormitories where the sick and injured slept. These snakes were introduced at the founding of each new temple of Asclepius throughout the classical world.
The original Hippocratic Oath began with the invocation "I swear by Apollo the Physician and by Asclepius and by Hygieia and Panacea and by all the gods ...".
Some later religious movements claimed links to Asclepius. In the 2nd century AD the controversial miracle-worker Alexander claimed that his god Glycon, a snake with a "head of linen" was an incarnation of Asclepius. The Greek language rhetorician and satirist Lucian produced the work Alexander the False Prophet to denounce the swindler for future generations. He described Alexander as having a character "made up of lying, trickery, perjury, and malice; [it was] facile, audacious, venturesome, diligent in the execution of its schemes, plausible, convincing, masking as good, and wearing an appearance absolutely opposite to its purpose." In Rome, the College of Aesculapius and Hygia was an association (collegium) that served as a burial society and dining club that also participated in the Imperial cult.
The botanical genus Asclepias (commonly known as milkweed) is named after him and includes the medicinal plant A. tuberosa or "Pleurisy root".
Asclepius was depicted on the reverse of the Greek 10,000 drachmas banknote of 1995–2001.
At the city of Miletus, archaeologists discovered a cave, under the city's theatre, which was associated with Asclepius cult.
The Bethesda Pool cult thing wasn't because there was a temple.
The history of the pool began in the 8th century BC, when a dam was built across the short Beth Zeta valley, turning it into a reservoir for rain water; a sluice-gate in the dam allowed the height to be controlled, and a rock-cut channel brought a steady stream of water from the reservoir into the city. The reservoir became known as the Upper Pool (בריכה העליונה).
In the 1st century BC, natural caves to the east of the two pools were turned into small baths, as part of an asclepieion; however, the Mishnah implies that at least one of these new pools was sacred to Fortuna, the goddess of fortune, rather than Asclepius, the god of healing.
There are places that are just simple natural places with mystical properties (according to the believers in these cults).
Others are the sacreed grooves.
A sacred grove or sacred woods are any grove of trees that are of special religious importance to a particular culture. Sacred groves feature in various cultures throughout the world. They were important features of the mythological landscape and cult practice of Celtic, Estonian, Baltic, Germanic, ancient Greek, Near Eastern, Roman, and Slavic polytheism, and continue to occur in locations such as India, Japan, and West Africa. Examples of sacred groves include the Greco-Roman temenos, various Germanic words for sacred groves, and the Celtic nemeton, which was largely but not exclusively associated with Druidic practice. During the Northern Crusades, there was a common practice of building churches on the sites of sacred groves. The Lakota and various other North American tribes consider particular forests or other natural landmarks to be sacred. Singular trees which a community deems to hold religious significance are known as sacred trees.
Many small sanctuaries consisted simply of a temenos with an altar and no temple. It was almost always necessary to undergo purification before being able to enter it.
The Roman legions could not enter Rome without first purifying themselves of shed blood.