Recent work in the acoustic metamaterial literature has focused on the design of metasurfaces that are capable of absorbing sound almost perfectly in narrow frequency ranges by coupling resonant effects to visco-thermal damping within their microstructure. Understanding acoustic attenuation mechanisms in narrow, viscous-fluid-filled channels is of fundamental importance in such applications. Motivated by recent work on acoustic propagation in narrow, air-filled channels, a theoretical framework is presented that demonstrates the controlling mechanisms of acoustic propagation in arbitrary Newtonian fluids, focusing on attenuation in air and water. For rigid-walled channels, whose widths are on the order of Stokes's boundary layer thickness, attenuation in air at 10 kHz can be over 200 dB m-1; in water it is less than 37 dB m-1. However, in water, fluid-structure-interaction effects can increase attenuation dramatically to over 77 dB m-1 for a steel-walled channel, with a reduction in phase-speed approaching 70%. For rigid-walled channels, approximate analytical expressions for dispersion relations are presented that are in close agreement with exact solutions over a broad range of frequencies, revealing explicitly the relationship between complex phase-speed, frequency and channel width.