Are BLS labor productivity measures in current dollars or constant dollars?
It depends on which BLS numbers you look at

For example in Table 2 of http://www.bls.gov/opub/mlr/2013/article/industry-employment-and-output-projections-to-2022.htm it gives for manufacturing output:

Billions of chained 2005 dollars            Billions of dollars
      2002        2012                        2002        2012
    4,320.8     4,407.6                     3,848.3     5,604.8


and the first pair of numbers uses a constant dollar (more precisely a chain-weighted dollar) measure while the second pair uses a current dollar measure 

For general productivity measures, the BLS says for example at http://www.bls.gov/opub/hom/pdf/homch12.pdf 


  Time-series indexes of manufacturing labor productivity (output per hour) and unit labor costs are constructed from three  basic  aggregate  measures:  Total  real  output,  hours worked,  and  nominal  compensation. 


So output per hour aims at a real measure of productivity, while unit labor costs divide a nominal figure by a real figure to give something which combines real productivity and nominal wages