Keep tissues and a bottle of hand sanitizer (or just rubbing alcohol) in your bag and wipe the seat down first. There's also the (less-than-perfect) method of covering the seat with toilet paper.
And practice GOOOD handwashing with soap, using a paper towel to turn off the faucet and open the bathroom door.
Lastly, remember all those studies that have shown time after time that things like the the handle for the water fountain, light switches, cellphones, keyboards, door handles, and ice dispensed from ice machines usually have a LOT more germs than toilet seats. (I just realized that, while my point was to make you less stressed about public restrooms, I may have freaked you about about ice and light switches. Sorry.)
Hand sanitizer & tissues!!!