The Cl 1604 supercluster at z = 0.9 is one of a small handful of such structures discovered in the high-redshift universe and is the first target observed as part of the Observations of Redshift Evolution in Large Scale Environments (ORELSE) survey. To date, Cl 1604 is the largest structure mapped at z ~ 1, with the most constituent clusters and the largest number of spectroscopically confirmed member galaxies. In this paper we present the results of a spectroscopic campaign to create a three-dimensional map of Cl 1604 and to understand the contamination by foreground and background large-scale structures. Combining new Deep Imaging Multi-Object Spectrograph observations with previous data yields high-quality redshifts for 1138 extragalactic objects in a ~0.08 deg² region, 413 of which are supercluster members. We examine the complex three-dimensional structure of Cl 1604, providing velocity dispersions for eight of the member clusters and groups. Our extensive spectroscopic data set is used to examine potential biases in cluster velocity dispersion measurements in the presence of overlapping structures and filaments. We also discuss other structures found along the line of sight, including a filament at z = 0.6 and two serendipitously discovered groups at z ~ 1.2.