If you are employing lean software development practices, then examining the root cause of defects should be part of your development process. Defects are one of the wastes - defects that go undetected as development happens cost time, especially as either they go from activity to activity (requirements defects slipping through design or even into integration and acceptance testing activities) or go undetected as the complexity of the system increases (as new features are added). Extending some effort to understand why defects aren't being detected early within the process and making changes to improve defect detection in upstream activities is part of maintaining a lean organization. However, it's also important to note that the cost of performing a root cause analysis is weighed against the benefits - it may not be every defect that is examined, but perhaps as a result of one iteration, there were a large number of defects that can be categorized and analyzed in groups to improve future iterations.
As far as structured methods go, there isn't a single standard approach. It depends on the problem and the team. Some methodologies include Eight Disciplines, DMAIC, Plan-Do-Check-Act, and others. Using these methodologies, you may employ tools such as orthogonal defect classification, fault tree analysis, 5 Whys, Ishikawa (fishbone) diagrams, and nominal group technique, among others (see the Seven Basic Tools of Quality for examples, although these primarily come from manufacturing and industrial processes and may not all translate one-for-one to software development activities), to narrow down the scope of the problem, identify commonalities, and get to the root causes and possible solutions.
If you're looking for a common process that exists across organizations, there really isn't one. However, a lot of the methodologies and tools are common. They are just broken down or used slightly differently across organizations and teams. You ultimately need to determine what works best for you and your team, in terms of defining the problem, determining the best methods to investigate and identify root causes, developing solutions and verifying that they are correct, and ensuring that they are having a positive impact on the process.