According to a report from a "senior law enforcement official," The Boston Globe noted that Dzhokhar Tsarnaev confessed to planting the Boston marathon bombs before he was read his Miranda rights. However, this is not necessarily a problem for the prosecution.
Miranda establishes that statements made by a suspect in custody in response to interrogation are not admissible against the defendant in court unless the defendant has been properly Mirandized. But, according to the Globe report, authorities aren't sweating this detail -- they believe that witness testimony from the man carjacked by the Tsarnaev brothers will serve as ample evidence for guilt, even if the 19-year-old suspect's hospital bed confession is not admissible in court. Via the Globe:
Tsarnaev’s attorneys are certain to challenge the legal admissibility of those admissions, and other information he gave them, such as claiming that he and his brothers acted alone, and that his brother was radicalized in an extreme form of Islam in part because he opposed U.S. actions in Iraq and Afghanistan.
But in an interview with the Globe, a senior police official said authorities are not worried about the initial admission to authorities being thrown out, because they have a strong witness: the man who was abducted by the Tsarnaev brothers last Thursday night.
Police sources told the Globe that the carjack victim has told police that Dzhokhar Tsarnaev and his older brother, Tamerlan, pointed guns at him and, in an apparent effort to intimidate the victim and dissuade him from trying anything foolish, Tamerlan Tsarnaev told him, “We just killed a cop. We blew up the marathon. And now we’re going to New York. Don’t [expletive] with us.”