Is it normal to ask for more money?

I just received a new job offer. I am currently employed and think this job opportunity is amazing! I really really want this job that was given to me but I was only okay with the salary offer they provided. I was okay with it but being told by everyone that I should ask for more since I do have pretty good credentials and background to do really well at this job. I asked for 3000 more than what they offered. I asked on Friday and I didn't receive an answer by the end of they day so I am assuming they will get back with me on Monday but I am really really scared that they will think I am greedy and will just revoke my offer completely. I asked politely and withy explanations for why I deserved more. I also reaffirmed to them that I am extremely interested in the job. So I didn't do my counter offer with disrespect or uninterest. I really want this job and I am scared that maybe I am being greedy in this economy and they are going to revoke my offer now. I know they can also come back and say no we can't offer you three grand more. And if they did I would still take it but I am just scared they will revoke my offer completely. What do you guys think?

Everyone tells me it's normal to ask for more and that usually when they give an initial offer they always lowball thinking the person will comeback and ask for a little more. But I am just scared cause I have never done a counter offer before. Please give me advice. I don't want my offer to be revoked.