Dear Dawna,
Most Spanish speaking countries don't have so extreme winters. May be our friends from Spain or Chile and Argentina suffer from a season with a high rate of colds and have a name for that.
Otherwise, you may translate it, as Steven Willis suggested, as "temporada de resfriados", even though such term may not be a usual expression, so you would say: la llamada temporada de resfriados en los Estados Unidos.
Saludos,
Antonio