"No dice" is a phrase that's believed to have originated in the United States in the early 20th century. Since gambling with dice was illegal in most states, crapshooters went to great lengths to hide and/or dispose of the bad bones when the law showed up (there were reports of gamblers actually swallowing the dice to avoid arrest). Prosecutors and judges had to throw out illegal-gambling charges when there were "no dice" (evidence).
According to phrases.org, the earliest known reference to the term was printed in the Port Arthur [Texas] Daily News in April 1921: "It's 'no dice' when the bones can't be found, according to a local court decision. Six white men were arrested Tuesday by Officer W. D. Moore and charged with gaming with dice." At the trial, the men were acquitted due to the lack of evidence. So "no dice" meant no conviction.
Somewhere along the line, like many gambling terms, "no dice" insinuated itself into the general slang, to mean "an unacceptable alternative," or "an unfavorable result," or "a refusal to accept a proposition" -- in short, "nothing doing."
