Is the value of the dollar really falling?
No.

At least, the dollar is not falling where it matters. Though the dollar is becoming weaker compared to the Euro, but not with major (third world) trading partners like China. These countries fix their currencies to the dollar, thereby ensuring relatively stable money. It also ensures that that US will not close the trade deficit in the near future, because the currencies of these countries will decline with the dollar.

Daniel Gross also wrote about this in his Moneybox column at Slate