The Workshop on Designing Accountable Software Systems (DASS) was convened in November 2024 with support from the U.S. National Science Foundation to engage a wide range of current and future stakeholders from government, academia, and industry on the cross-disciplinary topic of accountability in software systems. Over two days, attendees engaged in a series of panels, invited talks, and breakout sessions covering: (1) the dimensions of accountability, including legal compliance as well as business and societal aspects and drivers; (2) a conceptual model of the various structures needed to realize accountability; (3) the sources of legal requirements that affect software; (4) the operationalization of legal requirements in software; (5) the requirements to preserve evidence needed to conduct investigations; and (6) a range of challenges and contextual factors beyond software that affect why some accountability structures succeed, while others fail. The workshop was conducted as a collaborative systematization of knowledge that culminated in several research directions. The findings include the importance of clarifying definitions and responsibilities within accountable organizations, which can affect whether those researching accountability are making assumptions that limit the generalizability of findings. Further research was also identified as needed to study the ways to improve the translation of accountability structures into the software design process while improving engagement with stakeholders, such as legislators, regulators, business executives and system developers. Finally, a key finding was the high demands that DASS-like research projects place on interdisciplinary teams: both in terms of team formation and sustainment, as well as, the specific demands of cross-disciplinary learning that covers both research methods, research dissemination, and career development.