It largely depends on your idea of what you mean by travel, and what you mean by dimensions.
If you mean physical travel, technology has not yet gotten there. As for non-physical travel, many people believe you can perceive some non-physical realms in a state of deep meditation.
The most common 'place' to travel is the Astral realm. The word Astral was initially used to describe the mental plane. This is important, as whether someone believes it is purely mental or purely spiritual, everything perceived in the astral is filtered through the tropes already in your own memory. That is, your experience will vary greatly from someone else's.
Astral projection can begin simply. A lot of people use deliberate travel (called pathwalking in some traditions) to guide their mind's eye beyond their imaginations, into a state where they can interact with things outside of their own conscience. This usually involves a deliberate visualization of walking along a path, whether in nature or inside some structure, and passing through, up, or down, some transition, to another place. This may become fully immersive, but more often than not (especially early on), it happens while still at least mostly aware of the physical state of one's body. It is why meditation is so important: Observing only breath until everything else drops away, leaves the mind open to experience itself and not its outside circumstances.
Back on track: Astral projection does not mandate an out-of-body experience, though it may involve one. There are many methods of achieving this, though none are guaranteed, and some very accomplished practitioners never experience this regardless of their years of attempts. In these cases, it is more like a lucid dream, in which the astral body moves independently, and the physical body is not experienced. However, it is more like the awareness travels but the spirit does not leave the body. Anything physical which causes a disruption basically wakes the person up from their trance. While it may take a moment to become reoriented to the surroundings, it's like a sudden disturbance from deep meditation.
