A modified nodal integral method (MNIM) for two-dimensional, time-dependent Navier-Stokes equations is extended to three dimensions. The nodal integral method is based on local transverse integrations over finite size cells that reduce each partial differential equation to a set of ordinary differential equations (ODEs). Solutions of these ODEs in each cell for the transverse-averaged dependent variables are then utilized to develop the difference schemes. The discrete variables are scalar velocities and pressure, averaged over the faces of bricklike cells. The development of the MNIM is different from the conventional nodal method in two ways: (a) it is Poisson-type pressure equation based and (b) the convection terms are retained on the left side of the transverse-integrated equations and thus contribute to the homogeneous part of the solution. The first feature leads to a set of symmetric transverse-integrated equations for all the velocities, and the second feature yields distributions of constant + linear + exponential form for the transverse-averaged velocities. The scheme is tested on three-dimensional lid-driven cavity problems in cube- and prism-shaped cavities. Results obtained using the MNIM on fairly coarse meshes are comparable with reference solutions obtained using much finer meshes.