Qualifications
To become a cytogenetic tech, you'll need to get a bachelor's degree in a medical technology or laboratory technology program. Some hospitals require a specific certification in cytogenetics; two of the top programs in the U.S. for this are at the University of Texas, Houston, and at UCLA.
Both offer their junior- and senior-level students an intensive cytogenetics program at their teaching hospitals that'll qualify a student to take the cytogenetic certification exam.
Another thing you'll need for that certification is a year of experience. If you're really on the ball, you may be able to work part-time during college at a hospital in a lower-level lab or tech position to gain the required experience. You don't have to do it all at once, though.
Many people take it one step at a time: they get their degrees, then get licensed as a lab tech, then work for a year, and then get their certification. Each state has different requirements, so be sure to check requirements where you live or want to live and work.
It may seem unfair to jump through hoops for four years in school, only to have to jump through more hoops for the right certification. Happily, there are plenty of opportunities in this field.