In your standard layer by layer method, you presumably start with a cross, i.e. the four edge pieces in one face. It is fairly easy to make sure that whenever you solve one of those four edges, the two adjacent edges are correctly oriented. In this way you will have 5 of the 6 centres correct. When solving the rest of the first two layers, those centres will stay intact.
Solving the last layer can be a bit trickier. Some of the algorithms that layer-by-layer systems use will disturb the already solved centres. However, if you method does not disturb them, then you should be able to solve the pieces by placing them correctly relative to the final centre. If you have a cube where one of the centres does not have visible orientation, then it is useful to let that be the final face so that you don't have to worry about it.
Alternatively, you can solve the cube normally and fix the centres at the end. There is one easy to understand way to twist two centres, provided you know the 6-spot pattern and the 4-spot pattern:
- Find a twisted centre and determine how it needs to be rotated.
- Rotate the face with that centre that amount (so the centre is correct but the pieces around it are wrong).
- Do a 6-spot or 4-spot pattern to replace that centre by another one that needs to be twisted.
- Rotate the face back the way it was.
- Undo the 6-spot or 4-spot pattern by exactly reversing the moves you used.
This allows you to twist any centre by any amount, as long as you twist a different centre the same amount in the other direction.
(You can shave off a few moves by not doing a full 6-spot or 4-spot, but only doing just enough to replace the centre of one face by another.)
Twisting a single face centre is trickier, and I'm afraid I will have to use letters for the moves. A single centre can only be twisted by a half turn. Basically you have to give that face a half turn and then swap opposite edge pieces and opposite corner pieces in that layer.
Consider this move sequence:
R L U2 R' L' U2
Let's call this sequence
X. In the top layer (the U layer)
X swaps one pair of opposite corners, and one pair of opposite edges. The other two corners and two edges stay in place. The rest of the cube is mixed a bit.
Now consider what happens when you do
X U X' U'.
X swaps two edges and corners, the
U replaces them by the unswapped corners/edges, the
X' then swaps those as well (while also restoring the rest of the cube), and then
U' puts the top layer back again. So this puts all the pieces of the top layer in the opposite place, without twisting the centre. By following this with a
U2, everything is back in place and only the centre has twisted.
There are a few move cancellations:
X U X' U' U2 = (R L U2 R' L' U2) U (R L U2 R' L' U2)' U
= (R L U2 R' L' U2) U (U2 L R U2 L' R') U
= R L U2 R' L' U R L U2 R' L' U
= (R L U2 R' L' U)2