In fact, you don't need necessarily a credit card to sign up. You can sign up for the free trial by providing a bank account instead of a credit card. To verify that you own the account, you'll provide the account number and routing number. Google deposits a small random amount within 2-3 days. After you've received the deposit, you'll use the deposit amount to finish the process of signing up for the free trial. More details are to be read on the "Frequently Asked Questions"
page, at the level of the "Why do I need to provide a credit card or bank account for the free trial?" sub-chapter.