We call them "swingin' bridges" around here to denote a foot bridge as opposed to the big highway bridges. They are generally not big or fancy, just an inexpensive way to cross a wide span. The two most common designs have the walking deck either on top of the two cables, or hung from stringers below the cables. In either case, the decking has the same, or nearly the same droop as the supporting cables.
I invite you on a
pictoral history
of the bridges here, including the construction of the new one after the Flood of 97.
Then, if you are interested, see some of the
design considerations and math
that went into the newest bridge.