If you have an RN, do you have a year's experience in nursing? With the new CMS guidlines mandating charge RN's have at least 1 year nursing experience, this could be part of the problem. Many smaller units only have 1 RN on site, so they have to be more picky.
Since you asked about how people started: I started in a hospital that had in-house outpatient dialysis. I was part of the float (technician) pool and was asked to float to dialysis, which I did. I finished my nursing degree and stayed in dialysis. 3 years later I moved into acutes and never looked back
(I'm at 19 years of dialysis now, and run the acute program.)
There are other areas of nursing that you might be able to get into dialysis. We have a state run psych center that offers in house dialysis, as does the local prison. Not my cup of tea, but I have heard many dialysis nurses tell me it is very interesting. Also some rehab centers have their own in-house acute teams. You could also look into one of the groups that offers nursing home dialysis. In my area it is Affilated Dialysis and I believe they service 20-25 nursing homes.