You can minimize their appearance, though. Since the speakers are close to the wall, route the wires vertically behind the speakers so the speakers hid the wires as much as possible. If the left edge of the right speaker was closer to the right edge of the monitor, you could run the monitor wires horizontally behind the monitor until they were behind the speaker, then drop them down behind the speaker. That batch of wires on the right side of the desk can be run down the wall back behind the desk, then run along the wall to the right of the desk but under the desk instead of on top like it is now.
If the desk is built in, you could drill holes big enough for the cable connectors in the desktop by the wall to run the wires through. The glass top could then push up against the wires, minimizing the overhang of the glass at the front of the desk.
If you are not allowed to drill holes in the desk (or just don't want to because, say, it's an antique or a family heirloom or your renting), get some Wiremold, paint it the color of the wall, then stick it to wall using removable adhesive Command strips (make sure the wall is squeaky clean) and hide the wires inside the Wiremold.