We present a detailed series of experiments using spatially resolved flow velocimetry to examine the flow of Newtonian fluids through rectangular channels with one wavy surface of wave number k. The glass channels are fabricated by the method of selective laser-induced etching, which allows them to be made with a high (quasi-2D) aspect ratio (width/depth, w/2d=5) and with an accurate wave profile of small relative amplitude (A/d=0.05,A<k). Following the prior theoretical work for plane Couette flow over a wavy surface [Charru and Hinch, J. Fluid Mech. 414, 195 (2000)], we examine the influence of two dimensionless parameters (the normalized channel half-depth α=kd and the viscous length scale θ) on the penetration depth P of the perturbations that the wavy surface induces to the Poiseuille base flow. The asymptotic analysis by Charru and Hinch predicted three regimes of behavior classified as “shallow viscous” (P≈α), “deep viscous” (P≈1), and “inviscid” (P∼θ). All three regimes are here verified experimentally. Minor differences in details of the “phase diagram” for the flow regimes in α−θ parameter space observed between Poiseuille and plane Couette flow are attributed to the contrasting boundary conditions in the different flow configurations. Our experimental results also compare favorably to results from linear theory for a Poiseuille base flow and thus establish a detailed experimental complement to the theory.