Well, there is indeed a stereotypical "Samurai way of talking" that you can see in Samurai films or in historical dramas (時代劇, Jidaigeki) on TV, but it's far from being authentic. In fact, Samurai talked in many different ways, depending on the era and their home province (after all, they were speaking in their dialect).
As far as I know, the stereotypical Samurai speech in Jidaigeki is actually based on the Edo dialect of late Edo period. Many of the mannerisms you'd find in this speech do not specifically represent Samurai, but rather a typical resident of Edo in that particular time.
The most striking feature of this speech is the complete absence of the modern ～ます forms, which are sometimes replaced by other polite forms, but very often find Keigo used with plain forms. The most noticeable alternative polite form of this Edo-jidai speech is probably the polite/humble copula で御座る which is often used wholesale instead of any other copula. Just note that is copula is considered humble, so usually when speaking about someone else (at least someone you'd want to respectful to :)), you'd use the honorific copula でいらっしゃる instead of で御座る. For instance:
But when asking someone else for their name:
Note the first-person pronoun I used in the first example. 拙者 (せっしゃ) also highly identified with Samurai speech. It literally means something like "clumsy person", so it's a humble pronoun of course. In Jidaigeki, some Samurai use it, but the more haughty ones would probably use a different pronoun, such as おれ。
Another personal pronoun that's highly identified with Samurai speech is the second person pronoun お主. Again, you won't see every Jidaigeki Samurai using it, but it's highly stereotypical.