This is a popular claim - so much so that it has made its way into fictional portrayals of space travel. See, for example, the film Contact (1997), in which Jodie Foster's character is handed a suicide pill before leaving Earth to commune with aliens. This factoid was included in the film largely at the insistence of author Carl Sagan, on whose novel the film was based and who insisted that it was true. I am not certain where Carl Sagan got his authority to speak on the matter.

Basically everyone affiliated with the American space program who has addressed the claim has denied it. The technical adviser to Contact, a former NASA flight director, denied it. Jim Lovell, the commander of Apollo 13, denied it in his book Lost Moon (1994). I'm not aware of any credible evidence that would give us a reason not to believe these denials.

On the Soviet side: Alexey Leonov, the first man ever to conduct an extravehicular activity (EVA) in space (in 1965), allegedly said that he carried a suicide pill in the event that he was unable to re-enter his spacecraft, Voskhod 2. This appeared in Walking to Olympus: An EVA Chronology (PDF link) by David S. F. Portree and Robert C. Treviño of the NASA History Office (1997). The claim comes in a section with an ample list of cited sources, but it's not clear which one pertains to Leonov's alleged statement, so it's hard to confirm that he ever actually made the statement at all. Portree and Treviño could simply have been mistaken. And, of course, even if Leonov did make the statement, he could have been lying, exaggerating, or mis-remembering the event.

As some, like Lovell, have pointed out, giving space travelers suicide pills seems like an odd waste of cyanide. If astronauts want to die, they need only open their spacesuits or spacecraft to the void; death follows certainly and quickly.