Direct construction of Lebesgue measure I have seen two books for measure  theory, viz, Rudin's, and Lieb and Loss, "Analysis".
Both use some kind of Riesz representation theorem machinery to construct Lebesgue measure.
Is there a more "direct" construction, and if so, what is a source?
 A: The answer is, of course, yes. The other usual construction of the Lebesgue measure starts with the concept of outer measure. Then you introduce Caratheodory's definition of measurable set, and you develop the whole theory.
This approach is presented in a clean way in Royden's book, Real analysis.
While the approach based on Riesz' representation theorem is good for the purposes of functional analysis and PDE theory, the second approach is useful for geometric measure theory, also called "italian measure theory". Moreover, Rudin's definition is rather abstract in nature, and students usually do not understand that the underlying idea is that of covering sets with intervals and summing up their lengths. 
A: The most popular way is constructing it using the Caratheodory extension theorem, from Lebesgue outer measure. This approach is not very intuitive, but is a very powerful and general way for constructing measures.
An even more direct construction and essentially the one developed by Lebesgue himself defines Lebesgue measurable sets to be the ones that can be well approximated (in terms of outer measure) by open sets from the outside and by closed sets from the inside and shows that Lebesgue outer measure applied to these sets is an actual measure. You find this approach in A Radical Approach to Lebesgue's Theory of Integration by Bressoud and Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Stein and Shakarchi. 
