Jordan has an extensive Jewish history. From the time when the Jews conquered the land in Biblical times, it was owned by the tribes of Reuven, Gad, and Menashe. Unfortunately, little remains of that history in visible terms. The most “Jewish” sites you can see are the Kinneret, which you can view from the northern Roman historical site of Umm Qais, and the Dead Sea, which you have to be careful not to float across, or you’ll end up in Israel.
Jordan is probably one of the safest Arab countries a Jew can visit. The people there are overwhelmingly friendly, open, and welcoming. Most are fairly tolerant, although we did not advertise the fact that we are Jewish. Most people simply assumed we were Christian because Rabbi Ben told them we are from Canada and they do not realize that Jews live there, too. Most of the time, Rabbi Ben wore a hat instead of a kippah (although there are “Muslim” caps that look just like large sephardi yarmulkes – if you are uncomfortable about wearing a hat, you should buy and wear one of these instead). However, even those few people who did discover that we are Jewish did not bear us any ill will. We would not recommend overt Jewish observance there, although we did meet other Jews traveling there and we actually lit Chanukah menorah for the first night in the Amman airport.
As a warning, we have heard reports that border guards have refused to allow entry to Jews carrying tefillin. We have not been able to confirm this, as we had no problems crossing over into Aqaba from Eilat. Friends of ours have recommended splitting up the tefillin into two separate bags or packing them extremely well to avoid their detection.
Kosher food can be a challenge in Jordan. There are a lot of imports, but they are all from Arab countries. Although we looked, we were unable to find any packaged food imported from Western countries and bearing any sort of hecksher. However, there are markets nearly everywhere where you can buy fresh fruits and vegetables. If you will eat pas palter, you can buy pita bread at any number of bakeries that produce only pita bread en masse. For about 1 shekel’s worth (about 25 cents US) you can get as many as 6 or 7 fresh pitas. Because Jordan is right next door to Israel, it may be worth your while to stock up on food in Israel before traveling to Jordan. Our guesthouse in Irbid also allowed us to kasher some of their pots and cook in their kitchen.