We study the scheduling problem for a multi-source single-server generate-at-will (GAW) status update system with sources having heterogeneous service times and weights, with the goal of minimizing the weighted sum age of information (AoI). In particular, we study \emph{age-agnostic} schedulers which rely only on the first two moments of the source service times and they are relatively easier to implement than their age-aware counterparts which make use of the actual realizations of the service times. In particular, we focus on age-agnostic cyclic schedulers with $O(1)$ runtime complexity where status updates from multiple sources are scheduled according to a fixed finite transmission pattern. We first develop an analytical method to obtain the exact average AoI of each source when a transmission pattern is given. Then, we derive the optimum transmission pattern in closed form for the specific case of two sources. For general number of sources, we propose a novel algorithm, called IS (Insertion Search), for constructing transmission patterns, and we show that IS is capable of producing the optimum pattern for two-source systems, and it outperforms other existing age-agnostic schemes, for the case of more than two sources. Numerical examples are presented to showcase the effectiveness of the proposed approach.