We present a fully automated AI-based system for intensive monitoring of cognitive symptoms of neurotoxicity that frequently appear as a result of immunotherapy of hematologic malignancies. Early manifestations of these symptoms are evident in the patient's speech in the form of mild aphasia and confusion and can be detected and effectively treated prior to onset of more serious and potentially life-threatening impairment. We have developed the Automated Neural Nursing Assistant (ANNA) system designed to conduct a brief cognitive assessment several times per day over the telephone for 5-14 days following infusion of the immunotherapy medication. ANNA uses a conversational agent based on a large language model to elicit spontaneous speech in a semi-structured dialogue, followed by a series of brief language-based neurocognitive tests. In this paper we share ANNA's design and implementation, results of a pilot functional evaluation study, and discuss technical and logistic challenges facing the introduction of this type of technology in clinical practice. A large-scale clinical evaluation of ANNA will be conducted in an observational study of patients undergoing immunotherapy at the University of Minnesota Masonic Cancer Center starting in the Fall 2023.