Is it possible that the recipient of a heart transplant would display some of the donor's personality traits, as if the heart has memories?
Although there is clearly no feasible mechanism for such a phenomenon, there is good evidence that transplant patients can believe in some sort of transference of qualities from the donor. See for example (my emphasis):

Inspector, Y. et al. (2004) Another Person's Heart: Magical and Rational Thinking in the Psychological Adaptation to Heart Transplantation. The Israel Journal of Psychiatry and Related Sciences 41: 161-73.
Abstract
The goal of this study was to examine heart transplant recipients' psychological adaptation to another person's heart, with particular emphasis on recipients' attitudes toward graft and donor.
Thirty-five male heart recipients were examined by: the Symptom Distress Checklist (revised) (SCL-90-R); the Depression Adjective Checklist (DACL); a Post-Traumatic Stress Disorder Questionnaire (PTSD-Q); a Heart Image Questionnaire (HIQ); and a Semi-Structured Interview (SSI), aimed at eliciting attitudes and fantasies regarding the transplanted heart.
All instruments indicated high levels of stress even several years after the transplant, but, simultaneously, 73% of recipients felt that acquiring a new heart had had a dramatic influence on their lives with a new appreciation of the preciousness of life and a shift of priorities, toward altruism and spirituality. Sixty percent returned to work after the transplant but some had to adapt to a changed attitude from those around them who regarded them as anything from mystical creatures to vulnerable or still-sick individuals. While all recipients possessed a scientific knowledge of the anatomy and physiological significance of the heart (as revealed in the HIQ), many endorsed fantasies and displayed magical thinking: 46% of the recipients had fantasies about the donor's physical vigor and prowess, 40% expressed some guilt regarding the death of the donor, 34% entertained the possibility of acquiring qualities of the donor via the new heart. When asked to choose a most and least preferred imagined donor, 49% constructed their choices according to prejudices, desires, or fears related to ethnic, racial or sexual traits attributed to the donor.
This study confirms the intuitive idea that heart transplant involves a stressful course of events that produces an amplified sense of the precariousness of existence. Simultaneously, it gives rise to rejoicing at having been granted a new lease on life and a clear sense of new priorities, especially with regard to relationships. Less expectedly, this study shows that, despite sophisticated knowledge of anatomy and physiology, almost half the heart recipients had an overt or covert notion of potentially acquiring some of the donor's personality characteristics along with the heart. The concomitance of the magical and the logical is not uncommon in many areas of human existence, and is probably enhanced by the symbolic nature of the heart, and maybe, also, by the persistent stress that requires an ongoing, emotionally intense, adaptation process.

Another study is reported in:
Bunzel B, et al. (1992) Does changing the heart mean changing personality? A retrospective inquiry on 47 heart transplant patients. Qual Life Res. 1:251-6.
To actually investigate this further I suppose it would be necessary to set up experiments in which patients had transplant operations in which (unknown to the patient) no transplant actually took place, with the expectation that these individuals would display the same magical thinking. Grant application?