It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive characteristic field? What are the main difficulties in this setting; can one prove certain weaker versions of the main results of the theory (for example, using alterations instead of Hironaka's resolution of singularities)? Is there any text that treats these questions systematically?

1$\begingroup$ Have you looked at Takehiko Yasuda's papers? I haven't got time to look properly but they seem relevant. $\endgroup$ – Balazs Mar 5 '15 at 16:03
There are many theories of motivic integration. The first one, due to Kontsevich (on smooth varieties) and developed by DenefLoeser (in general, Inventiones Math., 1999) assumed originally that the characteristic was zero. Replacing arc schemes by Greenberg schemes, it has been generalized by Looijenga (Bourbaki seminar, Astérisque 276, 2002) to encompass varieties over $k[[t]]$, and then by Sebag (Bulletin SMF, 2004) to allow formal schemes over an arbitrary complete discrete valuation ring. A nice application is the definition by LoeserSebag (Duke Math. J., 2003) of the motivic Serre invariant of a rigid analytic space.
This geometric theory is presented in this generality in a book in preparation by Nicaise, Sebag and myself.
More recently, Cluckers and Loeser developed a more general theory (Inventiones Math., 2008 and Annals of Math., 2010) over henselian discretely valued fields of residual characteristic zero (rings such as $k[[t]]$, where the characteristic of $k$ is zero). The theory of Hrushovski and Kazhdan has a similar limitation. Cluckers and Loeser have then extended their theory to arbitrary henselian discretely valued fields of characteristic zero (Crelle, 2013).
While algebraic geometry works well in arbitrary characteristic, model theoretic inputs (e.g., Pas's theorem) presently impose such restrictions on the characteristic.