Photoionization of acetylene by extreme ultraviolet light results in a stand-alone contribution from the outermost valence orbital, followed by well-separated photoelectron bands from deeper molecular orbitals. This makes acetylene an ideal candidate for probing the photoionization dynamics in polyatomic molecules free from the spectral congestion often arising after interaction with an attosecond pulse train. Here, using an angle-resolved attosecond interferometric technique, we extract the photoionization time delays for the outermost valence orbital in acetylene relative to an atomic target, namely argon. Compared to argon, the photoemission from the acetylene molecule is found to be advanced by almost 28 attoseconds. The strong variation of the relative photoionization time delays as a function of the photoemission angle was interpreted using an analytical model based on semiclassical approximations to be the interplay between different short-range potentials along and perpendicular to the molecular axis. Our results highlight the importance of using attosecond time-resolved measurements to probe the nonspherical nature of the molecular potential, even in the case of relatively small, linear systems.