I've been advised by my mum to avoid cold foods or drinks e.g. watermelon, ice-cream, icy pole, iced drinks, etc. I'm not quite sure why but I think it's to minimize/prevent headaches and upsetting the stomach.
My sister always says to avoid tea and coffee during period to reduce caffeine consumption and thus reduce probabilities of headaches and dehydration. I also try to avoid fried and greasy food to get reduced cramps.
Diet and PMT/period pain are intricately related all month, not just during your period. What you eat in the weeks leading up to your period will influence your PMT/period and the accompanying level of pain and suffering, or lack thereof. By the time you get your period, the damage is done so to speak.
But as per the other answers, if you experience pain it is best to avoid cold things during your period - hot drinks, hot water bottles or packs will help. Avoid caffeine, sugar, saturated fats, takeaway foods and highly processed foods - try brown rice and vegetables, nourishing soups, hot water and herbal teas.