The Arabian peninsula has played an important part in history, but how were the bedoin populations there able to sustain themselves? From the information I've gathered food seems so sparse ad rainfall so rare that living there would probably get you killed. And to my knowledge there are few or no major river systems that could support agriculture.
There are no major rivers but there is ground water coming from the mountains in the west. There are several places where ground water is accessible, eg. places close to Riyadh or Hufuf Oasis in Eastern Province which is one of the largest oasis in the world. That ground water also supplies Bahrain and Quatar.
There are small permanent rivers like at the restored valley in Old Riyadh or in Yemen or Oman. There is also a bit of a river in Tafuk in Eastern Province but that one is based on ground water.
There is also rain water rushing down the valleys in the mountains after rain. That water can be stored with dams and used to water fields later on. Historic dams and water channels can be seen in Yemen and Najaf is green also today.
Plus there is some information that there was more water in historic times. Also it could be that water tunnels were used as that system was extensively used in Persia.
Source: book "Riyadh Heart of Arabia" , plus I have lived there for some time and seen that mini river in Tafuk.