Hi DRot, I'm not sure about your specific field, but in my experience (non-medical biological sciences) usually the main issue in getting any research position is funding. The thing is that most labs you approach will not co-incidentally have funding available for a new postdoc. So the question is, are you looking for a pre-funded position, or are you going to apply for your own funding?
If pre-funded, you probably need to check advertised job listings, and ask around through contacts etc to find out about openings, and apply specifically for those. In this case you will have less control over the project and lab you can work with, as normally the project is already specified and linked with the funding.
To apply for funding, do some research into fellowship schemes you could be eligible to apply for. Most universities will help you with this, check with your postgrad office, research office, etc. Then once you have an idea of the schemes, their criteria and deadlines you can contact prospective labs with a more concrete plan, and ask if they would be willing to work with you in putting together your project proposal and funding application. This way you will probably get a better response rate.
However, I would ask your supervisors and other academics in your field for advice on this, as they will know the specifics related to your field, and the places you are looking at applying to. Hope this helps- good luck!