Recruitment agencies

I am currently working in a very small accountancy practice (as accounts senior), and would like to get into a larger firm, still working in an accounts related job, but preferably not in an accountancy practice.

I very rarely see accounting jobs advertised other than through recruitment agents, so I have twice tried to find new employment through several agencies.  They have always said in the initial interview that they have plenty of jobs, of the type I would like, that they could put me forward for, and seem very positive towards me.  When it comes down to it however, they are only willing to put me forward for jobs that are exactly what I am trying to get out of - small accountancy practice jobs.  They do not seem willing to consider anyone without prior experience of exactly the same role.

Has everyone had the same experience with agencies?  If so, how do you get round this attitude?  If not, have I just gone to the wrong agencies or am I doing something wrong?

Thanks in advance for any help.