There are basically four choices with motion verbs in Japanese. Each has a slightly different implication.
に - "to" indicates the final goal of the travel. If chosen in your sentence it would be slightly nonsensical due to the この "Go directly to the street right here"
で - "in or around" tends to indicate meandering inside of the boundaries of a location. In other words, で treats the street not as a path to travel along, but as a place to move around inside of. This might be appropriate if you were at a street-fair or similar event (although this would probably be better with a verb other than 行く). In this case, however, the まっすぐ contradicts the implication of で, so it's not appropriate here. (EDIT: As several commentators have pointed out, で is a reasonable choice if there is a discussion of which way to travel. But this requires a context in which the conversants are discussing different options: "this road", "that sidewalk", "hang-gliding")
を - "across/along" indicates traveling the length (or a significant portion thereof) of the road/mountain/sky/(distance). More details are at this question
へ - "to" - very similar to に, and へ can be replaced with に in pretty much all situations involving movement verbs. There is a slight emphasis on the "direction" with へ, such that へ is often called the "direction particle", as opposed to the "destination particle" に.
There are, of course, other possible particles for marking destinations, such as まで, までで, までに, the catch-all topic marker は, and even simple omission.