An investigation of interior spacetimes sourced by stationary cylindrical anisotropic fluids is presented and specialized to rigidly rotating fluids with an azimuthally directed pressure. Based on the occurence of an extra degree of freedom in the equations, a general method for constructing different classes of exact solutions to the field equations has been found. Exemplifying such a recipe, a bunch of solutions are constructed. Axisymmetry and regularity conditions on the axis are examined and the spacetimes are properly matched to a vacuum exterior. A number of classes and subclasses are thus studied and an analysis of their features leads to sorting out two classes whose appropriate mathematical and physical properties are discussed. This work is part of a larger study of the influence of anisotropic pressure in GR, using cylindrical symmetry as a simplifying assumption, and considering in turn each principal stress direction. It has been initiated in companion Papers 1 and 2 where the pressure was assumed to be axially directed, and is followed by Paper 4 considering radial pressure and Paper 5 contrasting the previous results with analogous dust and perfect fluid solutions.