Here's one way I have used normal approximations in MCMC:
- Figure out what what would be a Gibbs sampler*, apart from say one conditional distribution that is tricky to sample but which will be approximately normal.
* (possibly with some variables integrated out of some of the conditionals - you can show this will still have the right stationary distribution etc, even though it's no longer a full conditional)
- using a normal approximation (usually one found algebraically, which makes things a lot faster, but it doesn't have to be) as the candidate distribution, perform a Metropolis-Hastings step on that variable. If the approximation is good you'll nearly always take the step. To perform the step you only need to be able to evaluate the tricky density at two known values, not generate from it. This is usually substantially easier.
There are some particular conditions you need to hold if you're not going to end up stuck somewhere (to do with the relative heaviness of the tails of the proposal and the actual conditional distribution).