We discuss the morphology and excitation of ionized gas in the nearby Sc galaxy NGC 3079. The almost edge-on orientation is ideal for studying the vertical structure of the gaseous disk, and especially the diffuse ionized medium (DIM) found between the bright H II regions. We used the Hawaii Imaging Fabry-Perot Interferometer (HIFI) to map 150,000 H alpha + [N II] lambda lambda 6548, 6583 emission-lime profiles across the entire disk, with resolution 70 km s(-1) at subarcsecond steps, down to a flux level of similar to 10(-17) ergs s(-1) cm(-2) (EM approximate to 4 cm(-6) pc). The DIM contributes similar to 30% of the total disk H alpha emission within a radius of 10 kpc. The DIM has broader emission lines and larger [N II]/H alpha flux ratios than the adjacent H II regions. Within a radius of 5 kpc, we find that the X-shaped filaments reported in previous studies emerge from the inner (R approximate to 1.5 kpc) disk, and rise more than 4 kpc above the disk plane. The morphology, kinematics, and excitation of the filaments suggest that they form a biconic interface between the undisturbed disk gas, and gas entrained in the wide-angle outflow. The DIM beyond 5 kpc radius is more vertically extended than the thick ionized disk detected in our Galaxy and in a few nearby edge-on systems. After correcting for dust, the vertical profile of this DIM has an exponential scale height of about 1.1 kpc, similar to that of the H I disk. The [N II] lambda 6583/H alpha flux ratio of the DIM increases monotonically with vertical height, reaching unity for \z\ greater than or similar to 2.5 kpc. The flux required to keep the DIM ionized at R = 8 kpc is similar to that near the solar circle of our Galaxy. Highly dilute radiation from O stars in the galactic plane probably maintains the DIM. The total mass of the DIM is of order 10(8)-10(9) M(.), representing less than 1% of the total dynamical mass of NGC 3079. Mechanical energy from intense star formation in the disk probably lifts the DIM above the disk. The several bubbles and filaments within 1 kpc of the disk plane is direct evidence for gas flow between the disk and halo.