In this work, we study unidirectional, fully developed, layered two-phase flows in microchannels, where the interface meets the wall at an arbitrary contact angle. Interfacial tension causes the interface to take the form of a circular arc, with a radius that depends on the contact angle. The momentum equations must therefore be solved on a domain with boundaries that are, in general, not iso-coordinate surfaces. We adopt the technique developed by Shankar (2005a), which extends the use of eigenfunctions to arbitrary shaped domains, and apply it to layered flows for rectangular and circular cross-sections of the channel. This method is computationally efficient and allows us to analyze in detail the effect of the contact angle on flow properties. We focus on the case of a rectangular channel, which is commonly encountered in microfluidic applications, and consider two distinct cases: (a) free interface whose equilibrium contact angle is a function of fluid wetting properties, and (b) pinned interface whose apparent contact angle is determined by fluid flow rates. We calculate the relationship between the volume fractions (holdups) and flow rate fractions of the fluids and show that a non-zero contact angle can significantly restrict the range of permissible flow rates. This range is greater when the less viscous fluid has a greater affinity for the wall. For fixed flow rates, the residence time of a fluid is found to increase as its affinity for the wall increases. The pressure drop, which directly impacts operational costs, is found to be lower when the more viscous fluid is more wetting. This non-intuitive result is explained in terms of the corresponding variation in fluid volume fractions. (C) 2018 Elsevier Ltd. All rights reserved.