In my experience, it is always been the ratio of products to reactants that mattered, so as long as your units are consistent; what they are doesn't matter. However, an in-class problem today required that the units be changed to molar (M) from millimolar (mM). Can you summarize when/why this matters in some cases, but not in others?
$K$ should be a unitless value. Otherwise, you'd get a nonsensical unit when you try to take the log of the unit. Each component of the product $K$ is usually an activity, which is frequently similar in value to the molar concentration, or a fugacity, which is frequently similar in value to the partial pressure of a gas.
If the equilibrium constant is defined as a ratio of quantities having units (concentrations, partial pressures, molalities), then unless the sum of the exponents in the numerator happens to equal the sum of the exponents in the denominator, the units matter.