Unfortunately, those types of pictures of electrons orbiting an atom are an old-fashioned and wrong picture of the electronic structure of an atom or molecule. With the development of quantum mechanics, we can attempt to visualize the electrons in an atom using atomic orbitals, which are a representation of the spatial density of the electron. As a quantum object, we can do experiments to measure the position of the electron at a specific time, but we cannot use a classical path to describe the behavior of the electron between measurements.
For molecules, we expect a similar description in terms of molecular orbitals. These can typically be approximated as combinations of atomic orbitals, which fits in very well with the interpretation that the atoms of a molecule "share" some of their electrons with one another. When a pair of electrons are shared between two atoms, we have a single covalent bond, when four electrons are shared a double bond, etc.
In considering the way atoms can be arranged to form a molecule, we have a competition between the relative attraction due to the formation of bonds and the relative repulsion due to electrostatic electron-electron and nucleus-nucleus repulsion. The more attractive the overall arrangment, the lower the potential energy and hence the more stable the resulting molecule will be. In theoretical chemistry, it is possible to compute molecular orbitals by adding up atomic orbitals and finding the particular sums that minimize the energy. This is a hard problem that almost always relies on numerical methods.
I'm not aware of any long-lived molecules that primarily share electrons between 3 atoms in an arrangement analogous to your picture. For example, cyclopropane is a triangular arrangement and there is a picture on that page showing the dominant orbital configuration, which involves pair-wise sharing. In terms of atomic orbitals there could be a small contribution where the electrons are shared by all 3 carbon atoms simulaneously, but it probably represents a higher energy configuration compared to the one pictured.
Actually, the cyclopropane geometry is already a bit strained, as you can also see on that diagram. The lobes of the atomic orbitals pictured there want to be straight. The bonding curves them around so that they can overlap, but this leads to a smaller binding energy than in ordinary, linear propane. The orbital that involves the 3-atom sharing is even more unnatural.
A case where it is easier to share electrons between more than 2 or 3 atoms is in aromatic compounds like benzene. There it is energetically favorable for the orbitals to "delocalize" as depicted in the 2nd figure in the Theory section on that page. This sort of delocalization that occurs most prominently in aromatic compounds is probably closest to the arrangement that you originally asked about.