Which statistical test should I use when there are multiple latent predictors and multiple latent outcomes? Which statistical method should I use when Predicting organization commitment (composed of 3 latent variables) using the five factors of personality?

Its a cross-sectional design with survey data. The Personality factors are Five independent factors (derived out of factor analysis) and each factor is measured on a likert scale, same goes for the three components of commitment and they are also measured on a likert type scale. 

The studies I have read upto now uses partial correlations and hierarchical regression to link these contructs but I somehow think there has to be a better way.

Note: this question was originally asked here on Stats.SE, but the moderator there suggested posting it on cogsci.se
I'm guessing you are relating the Big 5 factors of personality to Meyer and Allen's three factor model of commitment (normative, continuance, affective).

You certainly could adopt the simple approach of creating observed composite scores for each variable and reporting the correlation matrix and a set of multiple regressions.

However, the language of your question also suggests that you are interested in modelling latent variables rather than observed variables. This may suggest a structural equation modelling approach may be suitable.

Benefits of structural equation modelling approach


You can model your measurement model (i.e., do the observed variables load appropriately on your 8 latent factors; is there an overall higher order commitment factor that represents the three commitment factors)
You could model relationships between latent factors rather than observed variables (i.e., you can adjust for reliability of measurement); this often aligns analyses more with theoretical interests in psychology.
You could explore more sophisticated model comparison approaches. One approach I really like involves comparing models with various constraints in either the correlations or regression coefficients between the predictor set (i.e., personality) and the outcome set (i.e., commitment). E.g., one model could constrain all 15 regression coefficients between personality and commitment variables to be the same; another model could say that the relationships are constrained to be equal across commitment variables but different across personality variables.