Avoidance of culturally diverse peers undermines the benefits of multicultural teams. Using uncertainty reduction theory, we argue that team members’ cultural intelligence (CQ; the capability to work effectively in culturally diverse settings) and language competence (the ability to communicate in a language) exert main and synergistic effects on avoidance behaviors. We further propose that avoidance negatively predicts individuals’ task performance (behaviors that contribute to team outcomes) and voice behaviors (giving suggestions to improve the status quo). Put together, we test a moderated mediation model where the mediating role of avoidance between CQ and outcomes of task performance and voice behaviors is moderated by language competence. Our results, based on data collected from multicultural team members at three time points, showed no main effects of CQ and language competence on avoidance. Instead, we found synergistic effects such that CQ was negatively related to avoidance only when language competence was high. Avoidance was negatively related to peer‐rated task performance and voice behaviors. Conditional process analyses supported our moderated mediation hypothesis, suggesting that avoidance explains why people with higher CQ and higher language competence have higher task performance and voice behaviors.