A widely accepted view in the cultural evolutionary literature is that culture forms a dynamic system of elements (or 'traits') linked together by a variety of relationships. Despite this, large families of models within the cultural evolutionary literature tend to represent only a small number of traits, or traits without interrelationships. As such, these models may be unable to capture complex dynamics resulting from multiple interrelated traits. Here we put forward a systems approach to cultural evolutionary research-one that explicitly represents numerous cultural traits and their relationships to one another. Basing our discussion on simple graph-based models, we examine the implications of the systems approach in four domains: (i) the cultural evolution of decision rules ('filters') and their influence on the distribution of cultural traits in a population; (ii) the contingency and stochasticity of system trajectories through a structured state space; (iii) how trait interrelationships can modulate rates of cultural change; and (iv) how trait interrelationships can contribute to understandings of inter-group differences in realised traits. We suggest that the preliminary results presented here should inspire greater attention to the role of multiple interrelated traits on cultural evolution, and should motivate attempts to formalise the rich body of analyses and hypotheses within the humanities and social science literatures.