What could it be to make my Celica sputter then shut off after about 5-10 mins from being started?

I can start my car without much problem and then after about 5-10 minutes the car will sputter and then shut off. I cant start it again right after but I can start it again after like 20 or 30 mins and the same this happens again. I know its not the ecm because I replaced that. There is not a check engine light on and the lights and everything else work fine however long I have the key in. So its just the engine running is my only problem. What could it be and how can I fix it. Please no abreviations like MAF or IAC. I wont really know what those are. Also it doesnt idle rough but the idle goes up and down as if someone was reaving the engine up to about 2 grand for the first couple mins. Then after it quits doing that I know its getting ready to sputter and shut off. Please help. I love this car and dont want to get rid of it.