Let's go with an example. Three planets, all of roughly earth's size, follow the same orbit around a star with a mass of roughly 1 sun. Basically, all three planets are almost identical to Earth, orbiting a star almost identical to our sun, at an orbit almost identical to Earth's. How plausible is this, and in which ways would the planets interfere with each other, assuming they all orbit with the same velocity?
Something similar happens already with Jupiter and its Trojan asteroids:
The Jupiter trojans, commonly called Trojan asteroids or simply Trojans, are a large group of asteroids that share the planet Jupiter's orbit around the Sun. Relative to Jupiter, each Trojan librates around one of Jupiter's two stable Lagrange points: L4, lying 60° ahead of the planet in its orbit, and L5, 60° behind. Jupiter trojans are distributed in two elongated, curved regions around these Lagrangian points with an average semi-major axis of about 5.2 AU
Now, whether a large body like a planet would be stable around the L4 points is rather tricky: the L4 is a point, so anything outside it would be subjected to different forces and this, in the long term, could drive the system off balance.
The PlanetPlanet website has a section devoted to designing the ultimate solar system with the most habitable planets. Some of the solar systems designed are so improbable that one would have to believe that they would be constructed by super advanced civilziations instead of forming naturally.
The Ultimate Engineered Solar System packs 400 planets into the habitable zone around a star.
The design is based on recent calculations showing that planets can share the same orbit if there are at least seven planets in that orbit.
One way to have three objects and only three objects share the same orbit around a star would be to use a main object, a leading Trojan object, and a trailing Trojan object. But Trojan orbits require great differences in mass between the various bodies.
So if the leading Trojan object and the trailing Trojan object are both Earth sized planets, the main object orbiting between them would have to be a giant planet, or a dim star, or a brown dwarf intermediate in mass between a giant planet and a star.
So the middle object in the orbit could be a giant planet, brown dwarf, or tiny star, with an Earth sized moon or planet orbiting it, and two Earth sized planets in the leading Trojan and trailing Trojan positions.
See the Ultimate Trojan Solar System for a more elaborate Trojan solar system.
I think, you are looking for something like a Binary Planetary System. Now, we don't have much information on how planets typically form, look like, and interact, because we only know our solar system pretty well. However, we know that binary stars are quite common and even other more complex constellations exist. So at least it is imaginable that binary (or even tertiary) planets exist, that exhibit a similar dynamic. Imagine for example the Algol System, but replace Algol A and B with planets?
With the Trojans, it is another matter I think, because for that, the mass of the asteroid in the Lagrange point has to have "neglectable" mass, so this should only work for one very big and one small planet (basically a "faraway moon").
Here is another possible variant of what you want: https://en.m.wikipedia.org/wiki/Co-orbital_configuration#Co-orbital_moons
It has the advantage that it's more stable and works for planets of similar size, as well. Though probably not over billions of years. You can add a third planet for instance by exchanging of excentricity, if it is locked to the point where the two other planets exchange orbits. Or many other possibilities - all only semi-stable. In a very isolated solar system (far out at an arm of a galaxy or in the void between galaxies), the system could stay stable for longer.