I haven't worked very much with the new mesh functionality, so let me know if I've misunderstood your question.

(* Make a random Mesh to play with for this example*)
example = TriangulateMesh@DelaunayMesh[RandomReal[1, {50, 3}]]

(* Define a transformation matrix *)
myMat = {{1, 0, 0}, {2, 0, 1}, {0, 1, 0}};

(* Transform the mesh according to that matrix *)
TransformedRegion[example, myMat.{#1, #2, #3} &]

TransformedRegion can apparently take an arbitrary well defined function. So it's not just stuck with linear transforms.

If you want to work directly with the points defined inside of the mesh, that's possible as well but probably not necesssary. It can be done with http://reference.wolfram.com/language/FEMDocumentation/ref/ElementMesh.html