I think the main alternative to the Johansen (statistical) approach is the methodology propounded by Pesaran and Shin in their so-called long-run structural modelling (economic) approach. The main formal reference is Pesaran and Shin (2002). The methodology is also presented to a wider audience in Garratt et al. (2012).
Although referred to as the long-run structural modelling approach, you will also read about the type of models associated with the methodology, which are called VARX* models. The distinguishing feature is that the cointegrated models are estimated using reduced-rank regression and (over-)identifying restrictions are derived from economic a priori then tested. The sequencing is slightly different to Johansen since economic theory is given priority.
In the same school of thought is the cointegrated VAR approach associated with Juselius (2006). Again, Juselius propounds an economic approach to cointegration as opposed to the statistical approach by Johansen (and others like Phillips).
Pesaran's approach has been programmed and is available in the Microfit software.
Juselius' approach has also been programmed and is available in the RATS (CATS) software.
To my understanding, you can find MATLAB code in the GVAR toolbox, and this should give an indication of what's required for the Pesaran approach. In terms of programming, I haven't seen much difference when compared to Johansen (although I just quickly investigated this). Note that the GVAR approach can be thought of as an extension to the VARX approach (the difference lies in stacking individual country models, but estimation of the individual models is the same).
Pesaran, M and Shin, Yongcheol, (2002), LONG-RUN STRUCTURAL MODELLING, Econometric Reviews, 21, issue 1, p. 49-87.
Garratt, Anthony, Lee, Kevin, Pesaran, M and Shin, Yongcheol, (2012), Global and National Macroeconometric Modelling: A Long-Run Structural Approach, Oxford University Press.
Juselius, Katarina (2006) The Cointegrated VAR Model: Methodology and Applications (Advanced Texts in Econometrics).