This study is for patients who have been diagnosed with locoregionally advanced squamous cell carcinoma. The purpose of this study is to compare two usual treatment approaches to head and neck cancer: high-dose cisplatin given every 3 weeks with radiation to low-dose cisplatin given weekly with radiation. The usual approach for patients who are not in this study is treatment with radiation therapy combined with the chemotherapy drug cisplatin. Cisplatin is a drug approved by the Food and Drug Administration (FDA) to treat your head and neck cancer. Cisplatin can be given at different doses and at different times during radiation, but the most common way to give cisplatin is either as a high-dose every 3 weeks or a low-dose weekly during radiation. Participants can expect to be on this study for up to 7 weeks and then followed annually until disease progression.