For a zero order reaction, the unit of K is the unit of rate, i.e. mol dm-3 s-1
For a 1st order reaction, take the unit for a zero order reaction and subtract the unit of concentration i.e. (mol dm-3 s-1) - (mol dm-3) = s-1
For a 2nd order reaction, subtract another i.e. (s-1) - (mol dm-3) = mol-1 dm+3 s-1
etc.