An excellent 2005 review by Humphreys et al in JASN describes some of the most common causes of AKI in stem cell transplant patients.
The differential diagnosis of AKI in stem cell transplant recipients can be initially categorized by time period following transplant: that is, AKI developing in the immediate peri-transplant period, the "early period" (e.g., between 10-20 days post-transplant), and the "late period" (after a few months post-transplant).
The most common causes of immediate AKI are relatively rare, but include tumor lysis syndrome (which is fortunately becoming less common based on improved monitoring and newer drugs, such as Rasburicase) and toxic effects deriving from the marrow infusion itself--there is evidence that DMSO (an agent used to prolong cell life) can cause hemolysis and resultant AKI, for instance.
The most common causes of early AKI include standard causes (e.g., pre-renal failure from prolonged vomiting or diarrhea, obstruction, ATN from hypotension, etc) but also includes oncology-specific diagnoses such as vaso-occlusive disease (a subset of hepatorenal syndrome felt to be due to endothelial damage). These patients are often exposed to known nephrotoxins (e.g., amphotericin B, calcineurin inhibitors, iv contrast) which may also be playing a role. Hemorrhagic cystitis (a complication of high-dose Cytoxan often used as a conditioning regimen) can result in obstructive renal failure. Methotrexate, often used as an agent to prevent graft-versus host disease (GVHD), can cause a crystal-induced AKI. Patients with profound neutropenia are also at risk for infections (e.g., fungemia, bacterial sepsis) which frequently cause kidney injury.
In terms of late post-stem cell transplant causes of kidney failure, the main offender here is calcineurin inhibitor toxicity. In general, this affects individuals who get allogeneic stem cell transplants much more so than autologous transplants, as only the former generally require immunosuppressive agents such as CNIs in order to prevent GVHD, and helps explain why the rate of kidney injury is greater in allogeneic recipients compared with autologous ones. Many patients develop a low-grade thrombotic microangiopathy felt to be multifactorial in etiology which includes prior irradiation exposure causing endothelial damage, CNIs, and GVHD.