You need to use either a credit or debit card to order tickets through the site. If you don't have a card, you could obviously see if a friend or relative could order for you, and you could give them cash to cover the charge. Or another great option is the GREEN DOT Pay-As-You-Go VISA - which you can get at a bunch of different stores.
Having a "pay as you go" card like that is great for more than just ordering tickets. It works a lot like a "pay as you go" cell phone, and it only costs you $5/month for unlimited use! And you're only charged that fee when you have a balance on the card. If you stop loading it up, then they can't bill you anymore.
You just take the cash you want to load on the card to a participating store location and it gets put on the card for you to use online, at stores, or where ever you want (anywhere that accepts VISA).
**To see locations near you that have the GREEN DOT Pay-As-You-Go VISA, visit this URL: http://tinyurl.com/greendotlocations
Also, keep in mind that some artists elect to sell physical tickets to fans directly for cash. So depending on which artist(s) you know or are in contact with, getting tickets directly from them may be an option as well.