There are two types of grounds in an electrical system. You have the Equipment Grounding Conductor (EGC) and Grounding Electrode Conductor (GEC).
NEC Article 100 Definitions (Italicized text is my additional information):
Equipment Grounding Conductors (EGC) - The conductive path(s) that provides a ground-fault current path and connects normally non-current-carrying metal parts of equipment together and to the system grounded conductor or to the grounding electrode conductor, or both. This bonds the parts of the equipment that you do not expect to carry current under normal conditions (i.e., exposed metallic parts). This conductor is typically a bare copper conductor or has a green jacket or green jacket with a yellow stripe. This conductor is responsible for ensuring that all of the non-current-carrying parts of the equipment are of equal potential. This conductor does not carry current unless there is a fault condition. This is the "BONDING" conductor.
Grounding Electrode Conductor (GEC) - A conductor used to connect the system grounded conductor or the equipment to a grounding electrode or to a point on the grounding electrode system. This conductor is physically tied to earth (typically through a ground rod, but metallic plumbing direct buried in earth can be used, too). EGC are ultimately tied back to this point to provide a low-impedance path to ground in the event of a ground fault within the equipment. This is the "GROUNDING" conductor.
Three Phase Delta loads typically have only three phase conductors and a GEC that is somewhere near the three phase input terminal block. All EGCs within the equipment are tied back to this point as well.
Three Phase Wye loads typically have three phase conductors, one neutral, and one GEC. The Neutral is typically tied to the GEC and all EGCs are tied to this GEC. If a phase-to-phase fault occurs in the equipment, the neutral, which normally does not carry current under normal conditions, will suddenly carry current in the event of the fault; therefore, the Neutral is tied to the GEC and provides a path to ground that will not energize the exposed conductive surfaces of the equipment.
The picture you drew and provided above is called a "high leg delta" or "stinger." That type of connection is reserved for when you want to be able to tap a 240V single phase from a 480 three-phase service. Where I work, we do not permit high leg delta connections. On the rare installations where they exist, but have not been demolished yet, they are specially marked to alert the electricians of the potential danger.
I can't think of an installation where you would use a high leg delta on a purely three-phase load...in other words, I would not expect to see a three phase chiller pump connected to the source you drew above.
As a point of reference; some electrical systems have all six conductors! One example is a UPS-backed Power Distribution Unit, where the input feed from normal power would be a three phase wye source, complete with three Phase conductors, a Neutral conductor, a EGC conductor connecting the source panel and the UPS-backed PDU together, and a GEC tying the entire UPS-backed PDU and feeding panelboard to earth ground (Note that the panelboard is tied to earth through its own GEC as well). In effect, the source panel feeding the UPS-backed PDU and the UPS-backed PDU itself are both bonded together and solidly grounded.
All three phase systems will have a ground in one form or another (or both) - I think what you're asking is will all three phase systems have a grounded neutral. The answer is no. Purely three phase deltas do not have a grounded neutral, but high leg deltas and wye services do.