How exactly does one say, 'My name is," in Russian?

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Меня зовут Райан. (Menya zavoot - "o" sounds like "a").

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Menya zavoot name, which means "They call me name". I really don't like to read Russian put into our lettering. It is not that hard to learn Cyrillic. So it is 
Меня завут Борис. They call me Boris. Cyrillic is actually much more phonetic in spelling than English spelling, so it is a lot easier to sound out words, you just have to get used to a few strange looking letters.

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You can say also

Мое имя Райан.

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You can say : "Я Райан" (ya Ryan) - "I`m Ryan"

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My name is ...=  MinyA zavUt …   ( Меня зовут … )