Extracting elements from a matrix using a vector containing indices

I have a matrix of data that has a corresponding vector of indices.  I
would like to use those indices to extract specific matrix elements into a
new vector.  In other words, I have an R X C matrix with a corresponding
vector of C elements that have numbers mapping into specific elements of
the matrix.  I'd like to generate a new vector C long that contains those
elements.

My first approach is to iteratively grind through each matrix column to
manually extract the element corresponding to that column as referenced by
the appropriate index, and then appending that to a vector.  However, my R
instincts are that any time one has to craft a for loop to do something in
R that you are very likely Doing It Wrong; there's probably a very simple
one liner in R that will do all that that for loop would do.

I know there must be a simple "R way" to do this, but I remain flummoxed on
how to do so.  I've been feebly poking at the plyr package's maplyr() and
colwise() since I have a matrix and want to extract an array from it, and
this is inherently a column wise operation, so ... hmm.

Should I just give up and use a for loop?

M <- matrix(round(rnorm(20,20,10)),4,5)
M

Ix <- sample(1:4,5,replace=TRUE)
Ix

M[cbind(Ix,1:5)]