How to use mixed-effects models on multinomial data

Profesors Baron's and Bates' suggest that I use a linear mixed-effects model, and as 
a consequence, disregard the information that is contained in the ordering of my 6 
possible responses.

I don't think you "disregard order."  You simply count the number of
correct recalls, 0-5, and use that number as your dependent variable.
I don't see how that disregards order, unless you meant something else
by order, like the order in which the items were recalled.  What you
disregard is "ordinal regression".

 They further suggest that I plot the residuals against each of

my predictors.  This is to get an idea of how well the model fits the observed 
pattern of each of my predictors, right? If, say, for predictor1 the residuals are 
very large, that would mean that the model has fitted the pattern of this predictor 
very poorly, right?

I didn't mean "each predictor."  Rather, plot a graph of the residual
as a function of the predicted response, just as you would do with
ordinary regression.  (It is one of the default outputs for lm().)

 I have produced the lmer model and have tried to make the

residual plots, but have not succeeded. I can plot the residuals against the fitted 
values (but have to admit that I find it difficult to make sense of the plot), but 
how do I make separate plots for each of my predictors? Please let me know if I have 
misunderstood something here. 

Yes.  I think you have what you want.  If the residuals are in one
direction (high, or low) at one end or the other (left side or right
side), then your assumption that the response is predicted linearly
from the predictors is wrong.  You can also check for
homeoscedascicity.  (My spell checker chokes on this one no matter how
I spell it.)

Jon