Cross sectional data arranged in time series is referred to as panel data in econometrics. Analyzing panel data is a branch of econometrics all on its own. A good place to start is the Wikipedia article.
https://en.wikipedia.org/wiki/Panel_data
If I were given panel data to analyze I would consult William H. Greene's Econometric Analysis which has examples as well as a comprehensive overview of the theory.
Any Greene or Wooldridge textbook will certainly be a good place to start. It sounds like you are wanting to asses a treatment (assessing the impact of technology). There are some methods outlined in Greene or Wooldridge, but you may want to consider a matching technique. This is something I've recently discovered and I think it may be worth exploring if you're after the assessment of something.
Datla, consider going back to the fundamentals. Ask, what are the characteristics of this technology? What can we measure, and what are the surrogates? Then, ask what are characteristics of the phenomenon that the technology is trying to improve? Sketch a simple block diagram. At that point, import the physics of dynamical systems. Remember, the entire rational choice economics (and the tool of econometrics) attempts to capture the reality of dynamical systems. I urge you to read the grandfather of this work, Samuelson and his refuge in mathematical biology.
I know the answer is strange but it is better than the usual escape to run-of-the-mill publications on econometrics.
There is no best approach...because it depends! Each of the method's strength depends on what econometric problems you may try to account for while evaluating the impact of a certain technology in a panel data setting, e.g. endogeneity, sample selection, attrition, etc. The best way to understand these is to read some sections of Greene's Econometric Analysis and/or Wooldridge's Econometric Analysis of Cross-Section and Panel Data and/or Baltagi's Econometric Analysis of Panel Data. All these books are available online... Good Luck!!
In any given research study the analytical tool to be used depends on the objective and purpose of the study. Depending on the nature of the technology, and also time of evaluation i.e., pre or post impact assessment we have to decide on to use a qualitative or quantitative impact assessment tool. There are number of qualitative and quantitative analytical techniques available for that purpose. Further, ther I recommend to go through the following link for research article particularly on research impact assessment in agriculture using various approaches by Peter Weißhuhn Katharina Helming Johanna Ferretti published in Research Evaluation Volume 27, issue 1, pages 36-42.
In addition to the comments made so far, consider following two premises.
First, who are intended beneficiaries of the technology. If your cross section data covers the benefits accrued to them you may use any of the econometric tool.
Second, technology impact has implications on social dimensions. How far did you go on cross section data on qualitative aspects in the demand side. For this, you may have to consider any econometric tool capturing qualitative information on before and after or with and without type approaches.