We don't have a dishwasher, we have a double sink. I hate leaving dishes in there so one side is the dish drainer, and the other side is for dirty dishes.
That said - I have a little bit of counter space for a hand towel that I'll lay out any extra dishes that don't fit.
Do you use a special drying rack for bottles, nipples, pump parts etc? Would you just put them on a hand towel to dry?