Take a cab to Suweifiyah, a sort of upscale shopping district crammed with little boutiques and jewelry stores (also has the lion's share of the city's pubs and bars.) After exploring a bit (it's quite safe, and always bustling until well into the night), ask someone where to find Faroujna, which specializes in roast chicken and grilled meats- don't be put off by the modest downstairs, which has the kitchen and does take-away- take the spiral staircase upstairs, where the actual restaurant is. Other specialities include a particular large type of bread, and an amazing garlic dip which along with the usual assortment of pickled radishes/olvies, comes with anything you order. It's unhelpfully called "garlic" ("thum" in Arabic) on the menu, but it's amazingly creamy and addictive- sort of like a local version of aioli. Delish. Don't know how adventurous you are regarding offal, but I had some of the tastiest kidneys I've ever had there, along with some good chicken liver.
In the same general vein is a place called Mawal, near but not actually in Suweifiyah- try something called kebab halabi...
Other good places for cheap but good Jordanian food: Al-Quds diner downtown; the al-Jabri chain (try the mensaf), and the shawarma/juice places practically anywhere.
If you're looking for a more upscale night-out type place, Reem al-Bawadi is a tad hokey and overpriced, but still fun.
And if Arab food is not really your thing, check out the cafe at Wild Jordan in Jebel Amman (it's worth it for the view alone!) which has a very good brunch, and even does an afternoon tea. (Conveniently located walking distance from the turkish baths mentioned above- they're called the Pasha baths, and I second the recommendation!)
Yes, Petra is only about 5 hours from Amman if I remember correctly, so please go!
I didn't eat out much in Amman because I stayed with hosts, but I remember a place called Books@Cafe which is VERY popular with youngish types (not KIDS, but mid-20s to mid-30s). Tasty Western-style sandwiches, etc, and a bar as well. Sorry I don't know the address but like I said, it's popular...any cabbie or hotel clerk should know it.
Not a food tip, but there is also a Turkish bath near Books@Cafe. Because it is the only one in Amman (as far as I know) most people should be able to direct you. It's more expensive than some other baths in the region (outside Istanbul) at 15JD, but it's a really high-quality experience. Not quite as ...um...LOCAL-style as ones that cost only a few dollars. Good full-hour massage, good scrubbing, a jacuzzi and excellent steamroom, fluffy giant towels instead of pestamels...etc. It's great.
I often get stuck in Amman for days at a time.
There's a number of pretty good lebanese places - Bourj Al Hamam is one, but ask around. There's a great shwarma stand (biggest roll of meat I've ever seen) on the circle by the intercontinental. Cabs pull up, grab a couple of sandwiches for .25 JD, and drive off. Right across the street from the Intercon is a place called The Living Room which has decent sandwiches and sushi and a sort college/singles bar atmosphere, and a club upstairs, if you're coming out of Iraq and homesick for that. Don't EVER go to any of the hotel restaurants, especially not the Thai at the 4 seasons, except the sushi on the roof of the Howard Johnsons is good.
And yes, go to Petra. If you can convince your company, staying at the dead sea is about as close to the airport as downown, so you might be able to swing a couple of nights down there (the Marriot is better than the other one), but if you're paying, make it a stop on the way back from Petra.
And I've never done it but apparently you can fly from Makka airport, the little one that Airserve flies out of, to Akaba for about $50.
Sorry, more travel than food tips.
sigh...I just checked and see I never got around to filing my report on Amman, from my three weeks in Jordan last year...but in case you get out and see the country (and it would be shear insanity not to go to Petra for at least an overnight while you are there) check out these reports..:
And, unfortunatly, my Amman notes are temporarily unavailable due to a computer issue. IT says they are going to get them back, but in the meantime I have to operate from memory.
Anyway, while we ate in private homes in Amman a fair amount, here area a few suggestions:
Al Zawadeh was good for Lebanese/Middle Eastern.
Much better known, pricier, but also good, and in a great setting is Fakhr el-Din (link below). They have a huge menu so you should find something you like.
We had decent Italian at Romero's, across from the Intercontinental and very well-known. Service was lovely, as was the setting. Beware the wine list: they have some nice French (not as many Italian) selections, but they will cost you! (I ended up giving up on drinking wine in Jordan as everything was either bad or expensive, or both. French wines are your best bet for quality but not for value).
Rotisserie chicken places are ubiquitous in Amman in residential neighborhoods, and I never found a bad one.
Best option (IMO) if you get homesick is the Pizza Hut right across from the University of Jordan main gate. It actually isn't bad, and I am not a pizza fan(my daughter dragged me there). Please skip McDonalds. In general, however, the options in the University neighborhood are pretty skippable.
There is (or was last year anyway) a wonderful gelato place in the Rabyeh neighborhood behind the Days Inn (which is where we stayed), called Caspar's. (as in the friendly ghost). Worth checking out, and there is a decent shwerma place across the street from it with a nice patio (name escapes me). I found several good places just walking around that neighborhood, including the best flat bread I ever ate, fresh out of a wood oven at a bakery on a side street. The area isn't that far from Jabal Amman where you are staying, IIRC.
Inded, I recommend that you walk around and check out neighborhood places. Amman is a very safe city to walk in, and even at at night I never felt uncomfortable as a woman alone. My local friends assured me that it was indeed safe (though you will have to say no to endless taxi drivers who will pull up to offer you a ride) and I assume nothing has changed in the past year in that regard. It was on my walking adventures that I found the best restaurant eats (the best food in Jordan is in private homes, IMO), and it will help you feel less of that 'stuck-in-the-Hyatt' feeling, especially during a long stay!
Wish I could be more help, but I hope this gets you started!
do not forget the chinese resteraunt at the first circle. ask at the hotel to tell a driver to take you there, because it isn't exactly on the first circle. this is the first chinese resteraunt in amman, and has excellent arab style chinese food. get the chicken balls in special sauce.
I am sorry I don't remmeber names, but I would go to Shmisani neighborhood and find a plce for a hubbly bubble and some tea. also, ask at the hotel for a recomendation for a pastry shop - the pastry is great. go to the shouk and get coffee with cardomon to bring home, and syrian sweets.