Two suggestions:
(1) To sound courteous, use the modal verbs "could", "would", "might". They are less bossy than the more direct modals "can", "will", and "may".
(2) Instead of the accusatory "it's your turn", turn it into a question "whose turn?". Saying it this way avoids casting blame and produces the same outcome. Ask it sincerely, thereby leaving yourself vulnerable and treating yourselves as equals:
"Might (could, would) you dump the trash today?"
"The trash is full. Whose turn is it to dump it?"
Even better, you could say "Is it my turn to dump it?". That way you offer your own responsibility while leaving your roommate with an opportunity to take responsibility. Rather than confront your roommate with a chore, you offer an opportunity to show virtue.