Use rain barrel soaker hose, that will work with gravity. Be sure and use filter, 200 mesh at the least. Give your water some time to settle before running it through your soaker hose and don't have your outlet right at the bottom. You don't want all of the sediment going into your system, it will clog it.

Every 2.31 feet of elevation will give you 1 PSI. So if your barrels are 10' higher than your emitters, the pressure at the emitter will be 4.3 psi.
I would steer clear of the soaker hose and get some non-pressure compensating driptape such as Netafim Typhoon. Put an oversized 120 mesh disc filter at the bottom where the pressure is greatest and install 2 pressure gauges, one on each side of the filter. Clean the filter when there is a difference in pressure. Be sure to install a shuttoff valve before the filter so you can clean it without draining the tank.