What is the factual basis for "pirate speech"? (Did pirates really say things like "shiver me timbers"?)

There really isn't much of a basis in fact at all, but it has some non-fiction roots.

"Nearly all of our notions of their behavior come from the golden age of fictional piracy, which reached its zenith in 1881 with the appearance of Robert Louis Stevenson's Treasure Island." - Adams, C. "The Straight Dope", October 12, 2007 The Straight Dope – Fighting Ignorance Since 1973

So, According to Wikipedia, and several other sources. Our notions of pirates, and their dialects, is just a result of popularized fiction. Such as novels and movies like The Pirates of The Caribbean, Sinbad The Sailor, and Treasure Island.

-EDIT-

"So, was there a typical pirate accent at all? Among British outlaws, yes: The onboard speech was most likely underclass British sailor with extra curse words, augmented with a polyglot slang of French, Italian, Spanish, and Dutch picked up around the trade routes." -Christopher Bonanos

From: Slate

Summary: "Arrrg" is mainly fiction, but the accent could very well be a product of underclass European slang, and other languages picked up from around the world on trade routes.

A few answers here give good sources for various words that are commonly used when 'talking like a pirate'.

It may surprise some of you to learn though, that the 'accent' that most people go with (Rolled 'R's, dropped 'h's, gruff voice, etc.) actually originates from Robert Newton, the actor who played Long John Silver in the first sound production of Treasure Island.

So unfortunately, not a real pirate.

Sources: UK TV show, QI, which is rarely ever wrong and then follow up research that revealed Robert Newton as the 'Patron Saint' of Talk Like a Pirate Day for said reason.

In my experience, it seems that the dialect largely comes from two things

• Pirates are generally drunken sailors which gives birth to the 'Tavern Slur' style of speech
• Pirate songs!

Here's an excerpt from Lighthouse Journal

Music was, apparently, an important part of morale aboard any ship – pirate or otherwise. Often there would be a musician member aboard and tavern songs were popular with seamen in general, the concertina (‘squeeze box’) being the most popular on-board instrument. These songs, called chants or ‘Sea Shanty’, became part of the pirate lore. There were songs or sea shanties like:

• Capstan Shanty or Windlass Shanty – song to sing while raising the anchor of a ship.
• Short Drag Shanty – song sung while raising the masthead or trimming the sails.
• Halyard Shanty – song sung while raising the heavy sails from the yards, the wooden cross-pieces.
• Pumping Shanty – sung while pumping out the water when emptying the bilge.
• Forecastle Shanty – sung in the quarters of the crew members, the forecastle (fo’ksul) is the forward part of the main deck.
• Celebration Shanty – sung to celebrate anything worth celebrating, such as battle victories. The most known song is a tavern song called Blow the Man Down.

That last example's famous line, "Yo ho and blow the man down!" is a fairly good example.

Here are some others:

Here's to the grog, boys, the jolly, jolly grog
Here's to the rum and tobacco
I've a-spent all my tin with the lassies drinking gin
And to cross the briny ocean I must wander
  - from Here's to the Grog

To my, Ay, And we'll furl, Ay, And pay Paddy Doyle for his boots.
We'll sing, Ay, And we'll heave, Ay, And we'll hang Paddy Doyle for his boots.
  - from Patty Doyle

But now th' month is up, ol' turk. An' we say so, an' we hope so! Get up, ye swine, an' look for work. Oh! Poor old Man! Get up, ye swine, an' look for graft. An' we say so, an' we hope so! While we lays on an' yanks ye aft. Oh! Poor old Man!
  - from Dead Horse

Did roar, did roar, the crimps at me did roar.
There I went, me head all bent and the crimps at me did roar.
The first chap I ran afoul of was Mr. Shanghai Brown.
Well I asked him neat if he'd stand the treat; he looks me up and down.
He said "The last time yer was paid off you chalked me up no score.
But I'll give yes a chance and I'll take yer advance, and send yer to sea once more."
  - from Shanghai Brown

All in all I think the modern usage comes from a combination of songs like these, lots of drinking and the general usage of English and naval terms in the pirating age. Aye matey.