I would like to measure the impact of an intervention using a form of time series or some related technique (like segmented regression). I would like to use the Schools and Staffing Survey, which assesses my variable of interest over time, however, the respondents are different at each time point. I assume different respondents would present a problem for this kind of analysis. Can anyone suggest another technique that would allow me to gauge the impact of the intervention?
The data you have is sometimes called repeated cross section data. There are techniques that can be used with this kind of data. You can have a look at the Difference-in-Difference estimators (and their "matching" variants) presented by Blundell and Costa Dias (2000) as well as Smith and Todd (2005). There are some assumptions to be fulfilled. It's your role to check this and to argue that these estimators can be applied.
For a really complete picture, you can have a look at Heckman, Lalonde and Smith (2001).