Kevin, I'm going to give both you and Judaism the deserved respect; so this answer will be long but cannot claim to be complete and thorough. BTW, I tried fab4's links, but they didn't work for me.
Judaism has been around for about 4000 years. Issues of medical ethics, including organ transplantation are far from new. Halacha (Jewish law) is as deep and thorough as any legal or ethical investigation (ask any law professor with experience in both); but the goals and conclusions of halacha are aimed to fulfilling God's commandments as we have understood them for the last several thousand years.
Conclusions in halacha can change with the relevant data input. For instance, when heart transplants were first done the survival rate was pretty dismal. As such, halacha authorities (typically the few greatest rabbinic figures in a generation) were not quick to approve of this. As the risks have become smaller and the benefits larger, the halacha/law hasn't changed, but the conclusions have based on a different realitiy.
So any answer you get today, may be up for review in the future.
Lastly, the answer you get depends on who you ask. Voices in the more 'liberal' streams of Judaism (Reform, Conservative, etc.) are far more likely to give a relatively uncomplicated 'yes, go for it' to your question. The more traditional voices (Orthodox in the several variations, with which I indentify personally) are going to be more bound by the traditional definitions and applications and methodologies of halacha/Jewish law.
So, to your actual question: yes, maybe, no.
There are issues regarding both the donation and the receiving/transplantation. I won't try to cover them all. At the end of this answer I'll suggest some further sources for you. In general, there is not objection in principle to transplantation; the devil is in the details.
Judaism places a very high value on saving human life. That may not, however, usually come at the expense of another life. So, living donations and transplantations such as kidneys may be far less of a problem.
The problem arises in defining death. For general transplantation purposes in America, death is defined as brain stem death. There is a deabte among the greatest living Jewish authorities if that can be an acceptable definition in place of the traditional and established criteria of cardiac death used in Jewish law. If brain stem death is NOT acceptable, then according to the Torah if I remove an organ from a body with a beating heart, I am committing a murder. See the dilemna? So definition of death is of paramount importance and our biggest problem here. The HODS website has very good summary of this issue at http://www.hods.org/english/issuesE.shtml
. Look under Brain Stem Death.
When I raised this issue with our local transplant coordinator, she had never heard of it. I assure you that in your area (NYC) there are many Jews for whom this is very important. They will likely want to discuss this with their rabbi for guidance so as not to commit or be party to a grave transgression. There are Orthodox authorities who advocate relying on Brain Stem Death (BSD), and their congregants are more likely to donate. There are authorities who do not accept the BSD definition, and donation and transplantation are less likely for their communities. Again, I urge you to see the summary on the HODS website.
Fred Rosner, MD; Rabbi J. David Bleich, Rabbi Moshe Tendler are all authors in your area who have written on this in English. Dr. Rosner and Rabbi Bleich edited a volume called Jewish Bioethics which is now quite old, but an excellent introduction. The Schlesinger Institute at Shaarei Tzedek Hospital in Jerusalem, Israel published Jewish Medical Ethics (vol. I) with a more recent introduction to the topic. They also publish and English language edition of the Journal of Jewish Medical Ethics. Rabbi Bleich also covered some of this in his book from '81 title Judaism and Healing.
I hope this helps. For more info you might contact locally the students at Yeshiva University who are active in the National Medical Halacha Society.
mordechai y. scher